sum from n=0 to infinity of 5/(2+3^n)

{ "query": { "display": "$$\\sum_{n=0}^{\\infty\\:}\\frac{5}{2+3^{n}}$$", "symbolab_question": "BIG_OPERATOR#\\sum _{n=0}^{\\infty }\\frac{5}{2+3^{n}}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Series", "subTopic": "Convergence", "default": "\\mathrm{converges}" }, "steps": { "type": "interim", "title": "Check convergence of $$\\sum_{n=0}^{\\infty\\:}\\frac{5}{2+3^{n}}:{\\quad}$$converges", "input": "\\sum_{n=0}^{\\infty\\:}\\frac{5}{2+3^{n}}", "steps": [ { "type": "step", "primary": "Apply the constant multiplication rule: $$\\sum{c\\cdot{a_{n}}}=c\\cdot\\sum{a_{n}}$$", "result": "=5\\cdot\\:\\sum_{n=0}^{\\infty\\:}\\frac{1}{2+3^{n}}" }, { "type": "interim", "title": "Apply Series Ratio Test:$${\\quad}$$converges", "input": "\\sum_{n=0}^{\\infty\\:}\\frac{1}{2+3^{n}}", "steps": [ { "type": "definition", "title": "Series Ratio Test:", "text": "If there exists an $$N$$ so that for all $$n\\ge{N},\\:{\\quad}a_n\\neq{0}$$ and $$\\lim_{n\\to\\infty}|\\frac{a_{n+1}}{a_{n}}|=L:$$<br/>$${\\quad}$$If $$L<1$$, then $$\\sum{a_n}$$ converges<br/>$${\\quad}$$If $$L>1$$, then $$\\sum{a_n}$$ diverges<br/>$${\\quad}$$If $$L=1$$, then the test is inconclusive" }, { "type": "step", "primary": "$$\\left|\\frac{a_{n+1}}{a_n}\\right|=\\left|\\frac{\\frac{1}{2+3^{\\left(n+1\\right)}}}{\\frac{1}{2+3^{n}}}\\right|$$" }, { "type": "interim", "title": "Simplify $$\\left|\\frac{\\frac{1}{2+3^{\\left(n+1\\right)}}}{\\frac{1}{2+3^{n}}}\\right|:{\\quad}\\frac{\\left|2+3^{n}\\right|}{\\left|2+3^{n+1}\\right|}$$", "steps": [ { "type": "step", "result": "=\\left|\\frac{\\frac{1}{2+3^{n+1}}}{\\frac{1}{2+3^{n}}}\\right|" }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{1}{2+3^{n+1}}}{\\frac{1}{2+3^{n}}}:{\\quad}\\frac{2+3^{n}}{2+3^{n+1}}$$", "input": "\\frac{\\frac{1}{2+3^{n+1}}}{\\frac{1}{2+3^{n}}}", "result": "=\\left|\\frac{2+3^{n}}{2+3^{n+1}}\\right|", "steps": [ { "type": "step", "primary": "Divide fractions: $$\\frac{\\frac{a}{b}}{\\frac{c}{d}}=\\frac{a\\cdot\\:d}{b\\cdot\\:c}$$", "result": "=\\frac{1\\cdot\\:\\left(2+3^{n}\\right)}{\\left(2+3^{n+1}\\right)\\cdot\\:1}" }, { "type": "step", "primary": "Refine", "result": "=\\frac{2+3^{n}}{2+3^{n+1}}" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply absolute rule: $$\\left|\\frac{a}{b}\\right|\\:=\\frac{\\left|a\\right|}{\\left|b\\right|}$$", "result": "=\\frac{\\left|2+3^{n}\\right|}{\\left|2+3^{n+1}\\right|}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{\\left|2+3^{n}\\right|}{\\left|2+3^{n+1}\\right|}\\right)=\\frac{1}{3}$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{\\left|2+3^{n}\\right|}{\\left|2+3^{n+1}\\right|}\\right)", "steps": [ { "type": "step", "primary": "$$2+3^{n}$$ is positive when $$n\\to\\:\\infty\\:$$. Therefore $$\\left|2+3^{n}\\right|=2+3^{n}$$", "result": "=\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{2+3^{n}}{\\left|2+3^{n+1}\\right|}\\right)" }, { "type": "step", "primary": "$$2+3^{n+1}$$ is positive when $$n\\to\\:\\infty\\:$$. Therefore $$\\left|2+3^{n+1}\\right|=2+3^{n+1}$$", "result": "=\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{2+3^{n}}{2+3^{n+1}}\\right)" }, { "type": "interim", "title": "Divide by $$3^{n+1}:\\:\\frac{\\frac{2}{3^{n+1}}+\\frac{1}{3}}{\\frac{2}{3^{n+1}}+1}$$", "input": "\\frac{2+3^{n}}{2+3^{n+1}}", "result": "=\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{\\frac{2}{3^{n+1}}+\\frac{1}{3}}{\\frac{2}{3^{n+1}}+1}\\right)", "steps": [ { "type": "step", "primary": "Divide by $$3^{n+1}$$", "result": "=\\frac{\\frac{2}{3^{n+1}}+\\frac{3^{n}}{3^{n+1}}}{\\frac{2}{3^{n+1}}+\\frac{3^{n+1}}{3^{n+1}}}" }, { "type": "step", "primary": "Refine", "result": "=\\frac{\\frac{2}{3^{n+1}}+\\frac{1}{3}}{\\frac{2}{3^{n+1}}+1}" } ], "meta": { "interimType": "Generic Divide By 2Eq" } }, { "type": "step", "primary": "$$\\lim_{x\\to{a}}[\\frac{f\\left(x\\right)}{g\\left(x\\right)}]=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:\\quad\\lim_{x\\to{a}}{g\\left(x\\right)}\\neq0$$<br/>With the exception of indeterminate form", "result": "=\\frac{\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{2}{3^{n+1}}+\\frac{1}{3}\\right)}{\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{2}{3^{n+1}}+1\\right)}", "meta": { "title": { "extension": "Indeterminate Forms:<br/>$$\\frac{\\pm\\infty}{\\pm\\infty}$$<br/>$$\\frac{0}{0}$$<br/>$$\\pm\\infty\\cdot0$$<br/>$$0^0$$<br/>$$1^{\\pm\\infty}$$<br/>$$\\infty^{0}$$<br/>$$\\infty-\\infty$$" } } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{2}{3^{n+1}}+\\frac{1}{3}\\right)=\\frac{1}{3}$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{2}{3^{n+1}}+\\frac{1}{3}\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>With the exception of indeterminate form", "result": "=\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{2}{3^{n+1}}\\right)+\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{1}{3}\\right)", "meta": { "title": { "extension": "Indeterminate Forms:<br/>$$\\frac{\\pm\\infty}{\\pm\\infty}$$<br/>$$\\frac{0}{0}$$<br/>$$\\pm\\infty\\cdot0$$<br/>$$0^0$$<br/>$$1^{\\pm\\infty}$$<br/>$$\\infty^{0}$$<br/>$$\\infty-\\infty$$" } } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{2}{3^{n+1}}\\right)=0$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{2}{3^{n+1}}\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$", "result": "=2\\cdot\\:\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{1}{3^{n+1}}\\right)" }, { "type": "step", "primary": "$$\\lim_{x\\to{a}}[\\frac{f\\left(x\\right)}{g\\left(x\\right)}]=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:\\quad\\lim_{x\\to{a}}{g\\left(x\\right)}\\neq0$$<br/>With the exception of indeterminate form", "result": "=2\\cdot\\:\\frac{\\lim_{n\\to\\:\\infty\\:}\\left(1\\right)}{\\lim_{n\\to\\:\\infty\\:}\\left(3^{n+1}\\right)}", "meta": { "title": { "extension": "Indeterminate Forms:<br/>$$\\frac{\\pm\\infty}{\\pm\\infty}$$<br/>$$\\frac{0}{0}$$<br/>$$\\pm\\infty\\cdot0$$<br/>$$0^0$$<br/>$$1^{\\pm\\infty}$$<br/>$$\\infty^{0}$$<br/>$$\\infty-\\infty$$" } } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(1\\right)=1$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(1\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}{c}=c$$", "result": "=1" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(3^{n+1}\\right)=\\infty\\:$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(3^{n+1}\\right)", "steps": [ { "type": "interim", "title": "Apply exponent rules", "input": "3^{n+1}", "result": "=\\lim_{n\\to\\:\\infty\\:}\\left(e^{\\left(n+1\\right)\\ln\\left(3\\right)}\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{x}=e^{\\ln\\left(a^{x}\\right)}$$", "result": "=e^{\\ln\\left(3^{n+1}\\right)}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply log rule: $$\\ln\\left(a^{x}\\right)=x\\cdot\\ln\\left(a\\right)$$", "result": "=e^{\\left(n+1\\right)\\ln\\left(3\\right)}", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Apply Exp Rules 0Eq" } }, { "type": "interim", "title": "Apply the Limit Chain Rule:$${\\quad}\\infty\\:$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(e^{\\left(n+1\\right)\\ln\\left(3\\right)}\\right)", "steps": [ { "type": "definition", "title": "Limit Chain Rule", "text": "if $$\\lim_{u\\:\\to\\:b}\\:f\\left(u\\right)=L,\\:$$and $$\\lim_{x\\:\\to\\:a}g\\left(x\\right)=b,\\:$$and $$f\\left(x\\right)\\:$$is continuous at $$x=b$$<br/>$$\\quad$$Then: $$\\lim_{x\\:\\to\\:a}\\:f\\left(g\\left(x\\right)\\right)=L$$", "secondary": [ "$$g\\left(n\\right)=\\left(n+1\\right)\\ln\\left(3\\right),\\:f\\left(u\\right)=e^{u}$$" ] }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(\\left(n+1\\right)\\ln\\left(3\\right)\\right)=\\infty\\:$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(\\left(n+1\\right)\\ln\\left(3\\right)\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$", "result": "=\\ln\\left(3\\right)\\cdot\\:\\lim_{n\\to\\:\\infty\\:}\\left(n+1\\right)" }, { "type": "step", "primary": "$$\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>With the exception of indeterminate form", "result": "=\\ln\\left(3\\right)\\left(\\lim_{n\\to\\:\\infty\\:}\\left(n\\right)+\\lim_{n\\to\\:\\infty\\:}\\left(1\\right)\\right)", "meta": { "title": { "extension": "Indeterminate Forms:<br/>$$\\frac{\\pm\\infty}{\\pm\\infty}$$<br/>$$\\frac{0}{0}$$<br/>$$\\pm\\infty\\cdot0$$<br/>$$0^0$$<br/>$$1^{\\pm\\infty}$$<br/>$$\\infty^{0}$$<br/>$$\\infty-\\infty$$" } } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(n\\right)=\\infty\\:$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(n\\right)", "steps": [ { "type": "step", "primary": "Apply the common limit: $$\\lim_{n\\to\\:\\infty\\:}\\left(n\\right)=\\infty\\:$$", "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(1\\right)=1$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(1\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}{c}=c$$", "result": "=1" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=\\ln\\left(3\\right)\\left(\\infty\\:+1\\right)" }, { "type": "interim", "title": "Simplify $$\\ln\\left(3\\right)\\left(\\infty\\:+1\\right):{\\quad}\\infty\\:$$", "input": "\\ln\\left(3\\right)\\left(\\infty\\:+1\\right)", "result": "=\\infty\\:", "steps": [ { "type": "interim", "title": "$$\\infty\\:+1=\\infty\\:$$", "input": "\\infty\\:+1", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$\\infty\\:+c=\\infty\\:$$", "result": "=\\infty\\:" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\ln\\left(3\\right)\\left(\\infty\\:\\right)" }, { "type": "interim", "title": "$$\\ln\\left(3\\right)\\left(\\infty\\:\\right)=\\infty\\:$$", "input": "\\ln\\left(3\\right)\\left(\\infty\\:\\right)", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$c\\cdot\\infty=\\infty$$", "result": "=\\infty\\:" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Yh4w7EfCxZSg4ozeO+skXBgUlp2a3w1QZHYsaqXCjTxwkKGJWEPFPk38sdJMsyPIwhWwQYyckKycWgIwD4f3sPC30sSftAIFS6Qkpy19IkqxSGm+I7f73Ci1MIDs4v5cqT2ATb/SVwJa8gQiZNBNYLCI2sSeA74029n2yo277ZU=" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{u\\to\\:\\infty\\:}\\left(e^{u}\\right)=\\infty\\:$$", "input": "\\lim_{u\\to\\:\\infty\\:}\\left(e^{u}\\right)", "steps": [ { "type": "step", "primary": "Apply the common limit: $$\\lim_{u\\to\\:\\infty\\:}\\left(e^{u}\\right)=\\infty\\:$$", "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "primary": "By the chain rule:", "result": "=\\infty\\:" } ], "meta": { "interimType": "Limit Chain Rule 0Eq", "practiceLink": "/practice/limits-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Limit Chain Rule" } }, { "type": "step", "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=2\\cdot\\:\\frac{1}{\\infty\\:}" }, { "type": "interim", "title": "Simplify $$2\\cdot\\:\\frac{1}{\\infty\\:}:{\\quad}0$$", "input": "2\\cdot\\:\\frac{1}{\\infty\\:}", "result": "=0", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$\\frac{c}{\\infty}=0$$", "result": "=2\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvioOn9HZuLHQfol+FkTFV5erehkKrn0era9rz8TlL+x/vZuJKdCFsPJy1+5gBMEc9doEFMST8lDZxn1Yq5HMKVTuF2Uk9xnPPKEGoObx02AbXJCTae4I8X+gaGFYzQzhr2qjWqX1f6iAOqMbShkzd9w0=" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{1}{3}\\right)=\\frac{1}{3}$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{1}{3}\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}{c}=c$$", "result": "=\\frac{1}{3}" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=0+\\frac{1}{3}" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{1}{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{2}{3^{n+1}}+1\\right)=1$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{2}{3^{n+1}}+1\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>With the exception of indeterminate form", "result": "=\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{2}{3^{n+1}}\\right)+\\lim_{n\\to\\:\\infty\\:}\\left(1\\right)", "meta": { "title": { "extension": "Indeterminate Forms:<br/>$$\\frac{\\pm\\infty}{\\pm\\infty}$$<br/>$$\\frac{0}{0}$$<br/>$$\\pm\\infty\\cdot0$$<br/>$$0^0$$<br/>$$1^{\\pm\\infty}$$<br/>$$\\infty^{0}$$<br/>$$\\infty-\\infty$$" } } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{2}{3^{n+1}}\\right)=0$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{2}{3^{n+1}}\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$", "result": "=2\\cdot\\:\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{1}{3^{n+1}}\\right)" }, { "type": "step", "primary": "$$\\lim_{x\\to{a}}[\\frac{f\\left(x\\right)}{g\\left(x\\right)}]=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:\\quad\\lim_{x\\to{a}}{g\\left(x\\right)}\\neq0$$<br/>With the exception of indeterminate form", "result": "=2\\cdot\\:\\frac{\\lim_{n\\to\\:\\infty\\:}\\left(1\\right)}{\\lim_{n\\to\\:\\infty\\:}\\left(3^{n+1}\\right)}", "meta": { "title": { "extension": "Indeterminate Forms:<br/>$$\\frac{\\pm\\infty}{\\pm\\infty}$$<br/>$$\\frac{0}{0}$$<br/>$$\\pm\\infty\\cdot0$$<br/>$$0^0$$<br/>$$1^{\\pm\\infty}$$<br/>$$\\infty^{0}$$<br/>$$\\infty-\\infty$$" } } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(1\\right)=1$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(1\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}{c}=c$$", "result": "=1" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(3^{n+1}\\right)=\\infty\\:$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(3^{n+1}\\right)", "steps": [ { "type": "interim", "title": "Apply exponent rules", "input": "3^{n+1}", "result": "=\\lim_{n\\to\\:\\infty\\:}\\left(e^{\\left(n+1\\right)\\ln\\left(3\\right)}\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{x}=e^{\\ln\\left(a^{x}\\right)}$$", "result": "=e^{\\ln\\left(3^{n+1}\\right)}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply log rule: $$\\ln\\left(a^{x}\\right)=x\\cdot\\ln\\left(a\\right)$$", "result": "=e^{\\left(n+1\\right)\\ln\\left(3\\right)}", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Apply Exp Rules 0Eq" } }, { "type": "interim", "title": "Apply the Limit Chain Rule:$${\\quad}\\infty\\:$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(e^{\\left(n+1\\right)\\ln\\left(3\\right)}\\right)", "steps": [ { "type": "definition", "title": "Limit Chain Rule", "text": "if $$\\lim_{u\\:\\to\\:b}\\:f\\left(u\\right)=L,\\:$$and $$\\lim_{x\\:\\to\\:a}g\\left(x\\right)=b,\\:$$and $$f\\left(x\\right)\\:$$is continuous at $$x=b$$<br/>$$\\quad$$Then: $$\\lim_{x\\:\\to\\:a}\\:f\\left(g\\left(x\\right)\\right)=L$$", "secondary": [ "$$g\\left(n\\right)=\\left(n+1\\right)\\ln\\left(3\\right),\\:f\\left(u\\right)=e^{u}$$" ] }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(\\left(n+1\\right)\\ln\\left(3\\right)\\right)=\\infty\\:$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(\\left(n+1\\right)\\ln\\left(3\\right)\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$", "result": "=\\ln\\left(3\\right)\\cdot\\:\\lim_{n\\to\\:\\infty\\:}\\left(n+1\\right)" }, { "type": "step", "primary": "$$\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>With the exception of indeterminate form", "result": "=\\ln\\left(3\\right)\\left(\\lim_{n\\to\\:\\infty\\:}\\left(n\\right)+\\lim_{n\\to\\:\\infty\\:}\\left(1\\right)\\right)", "meta": { "title": { "extension": "Indeterminate Forms:<br/>$$\\frac{\\pm\\infty}{\\pm\\infty}$$<br/>$$\\frac{0}{0}$$<br/>$$\\pm\\infty\\cdot0$$<br/>$$0^0$$<br/>$$1^{\\pm\\infty}$$<br/>$$\\infty^{0}$$<br/>$$\\infty-\\infty$$" } } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(n\\right)=\\infty\\:$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(n\\right)", "steps": [ { "type": "step", "primary": "Apply the common limit: $$\\lim_{n\\to\\:\\infty\\:}\\left(n\\right)=\\infty\\:$$", "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(1\\right)=1$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(1\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}{c}=c$$", "result": "=1" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=\\ln\\left(3\\right)\\left(\\infty\\:+1\\right)" }, { "type": "interim", "title": "Simplify $$\\ln\\left(3\\right)\\left(\\infty\\:+1\\right):{\\quad}\\infty\\:$$", "input": "\\ln\\left(3\\right)\\left(\\infty\\:+1\\right)", "result": "=\\infty\\:", "steps": [ { "type": "interim", "title": "$$\\infty\\:+1=\\infty\\:$$", "input": "\\infty\\:+1", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$\\infty\\:+c=\\infty\\:$$", "result": "=\\infty\\:" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\ln\\left(3\\right)\\left(\\infty\\:\\right)" }, { "type": "interim", "title": "$$\\ln\\left(3\\right)\\left(\\infty\\:\\right)=\\infty\\:$$", "input": "\\ln\\left(3\\right)\\left(\\infty\\:\\right)", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$c\\cdot\\infty=\\infty$$", "result": "=\\infty\\:" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Yh4w7EfCxZSg4ozeO+skXBgUlp2a3w1QZHYsaqXCjTxwkKGJWEPFPk38sdJMsyPIwhWwQYyckKycWgIwD4f3sPC30sSftAIFS6Qkpy19IkqxSGm+I7f73Ci1MIDs4v5cqT2ATb/SVwJa8gQiZNBNYLCI2sSeA74029n2yo277ZU=" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{u\\to\\:\\infty\\:}\\left(e^{u}\\right)=\\infty\\:$$", "input": "\\lim_{u\\to\\:\\infty\\:}\\left(e^{u}\\right)", "steps": [ { "type": "step", "primary": "Apply the common limit: $$\\lim_{u\\to\\:\\infty\\:}\\left(e^{u}\\right)=\\infty\\:$$", "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "primary": "By the chain rule:", "result": "=\\infty\\:" } ], "meta": { "interimType": "Limit Chain Rule 0Eq", "practiceLink": "/practice/limits-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Limit Chain Rule" } }, { "type": "step", "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=2\\cdot\\:\\frac{1}{\\infty\\:}" }, { "type": "interim", "title": "Simplify $$2\\cdot\\:\\frac{1}{\\infty\\:}:{\\quad}0$$", "input": "2\\cdot\\:\\frac{1}{\\infty\\:}", "result": "=0", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$\\frac{c}{\\infty}=0$$", "result": "=2\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvioOn9HZuLHQfol+FkTFV5erehkKrn0era9rz8TlL+x/vZuJKdCFsPJy1+5gBMEc9doEFMST8lDZxn1Yq5HMKVTuF2Uk9xnPPKEGoObx02AbXJCTae4I8X+gaGFYzQzhr2qjWqX1f6iAOqMbShkzd9w0=" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(1\\right)=1$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(1\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}{c}=c$$", "result": "=1" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=0+1" }, { "type": "step", "primary": "Simplify", "result": "=1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=\\frac{\\frac{1}{3}}{1}" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{1}{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "primary": "$$L<1,\\:$$by the ratio test", "result": "=\\mathrm{converges}" } ], "meta": { "interimType": "Series Apply Ratio Test 0Eq" } }, { "type": "step", "result": "=5\\mathrm{converges}" }, { "type": "step", "result": "=\\mathrm{converges}" } ], "meta": { "solvingClass": "Series", "practiceLink": "/practice/series-practice#area=main&subtopic=Ratio%20Test", "practiceTopic": "Series Ratio Test" } } }