sum from n=3 to infinity of (3^n)/(11^n)

{ "query": { "display": "$$\\sum_{n=3}^{\\infty\\:}\\frac{3^{n}}{11^{n}}$$", "symbolab_question": "BIG_OPERATOR#\\sum _{n=3}^{\\infty }\\frac{3^{n}}{11^{n}}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Series", "subTopic": "Convergence", "default": "\\mathrm{converges}" }, "steps": { "type": "interim", "title": "Check convergence of $$\\sum_{n=3}^{\\infty\\:}\\frac{3^{n}}{11^{n}}:{\\quad}$$converges", "input": "\\sum_{n=3}^{\\infty\\:}\\frac{3^{n}}{11^{n}}", "steps": [ { "type": "interim", "title": "Apply Series Ratio Test:$${\\quad}$$converges", "input": "\\sum_{n=3}^{\\infty\\:}\\frac{3^{n}}{11^{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": "$$\\left|\\frac{a_{n+1}}{a_n}\\right|=\\left|\\frac{\\frac{3^{\\left(n+1\\right)}}{11^{\\left(n+1\\right)}}}{\\frac{3^{n}}{11^{n}}}\\right|$$" }, { "type": "interim", "title": "$$\\left|\\frac{\\frac{3^{\\left(n+1\\right)}}{11^{\\left(n+1\\right)}}}{\\frac{3^{n}}{11^{n}}}\\right|=\\frac{3}{11}$$", "steps": [ { "type": "step", "result": "=\\left|\\frac{\\frac{3^{n+1}}{11^{n+1}}}{\\frac{3^{n}}{11^{n}}}\\right|" }, { "type": "step", "primary": "Divide fractions: $$\\frac{\\frac{a}{b}}{\\frac{c}{d}}=\\frac{a\\cdot\\:d}{b\\cdot\\:c}$$", "result": "=\\left|\\frac{3^{n+1}\\cdot\\:11^{n}}{11^{n+1}\\cdot\\:3^{n}}\\right|" }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=x^{a-b}$$", "secondary": [ "$$\\frac{3^{n+1}}{3^{n}}=3^{n+1-n}$$" ], "result": "=\\frac{11^{n}\\cdot\\:3^{n-n+1}}{11^{n+1}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add similar elements: $$n+1-n=1$$", "result": "=\\frac{3\\cdot\\:11^{n}}{11^{n+1}}" }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=\\frac{1}{x^{b-a}}$$", "secondary": [ "$$\\frac{11^{n}}{11^{n+1}}=\\frac{1}{11^{n+1-n}}$$" ], "result": "=\\frac{3}{11^{n+1-n}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add similar elements: $$n+1-n=1$$", "result": "=\\left|\\frac{3}{11}\\right|" }, { "type": "step", "primary": "Apply absolute rule: $$\\left|a\\right|=a,\\:a\\ge0$$", "result": "=\\frac{3}{11}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver" } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{3}{11}\\right)=\\frac{3}{11}$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{3}{11}\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}{c}=c$$", "result": "=\\frac{3}{11}" } ], "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": "=\\mathrm{converges}" } ], "meta": { "solvingClass": "Series", "practiceLink": "/practice/series-practice#area=main&subtopic=Ratio%20Test", "practiceTopic": "Series Ratio Test" } } }