sum from n=0 to infinity of ne^{-n}

{ "query": { "display": "$$\\sum_{n=0}^{\\infty\\:}ne^{-n}$$", "symbolab_question": "BIG_OPERATOR#\\sum _{n=0}^{\\infty }ne^{-n}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Series", "subTopic": "Convergence", "default": "\\mathrm{converges}" }, "steps": { "type": "interim", "title": "Check convergence of $$\\sum_{n=0}^{\\infty\\:}ne^{-n}:{\\quad}$$converges", "input": "\\sum_{n=0}^{\\infty\\:}ne^{-n}", "steps": [ { "type": "interim", "title": "Apply Series Ratio Test:$${\\quad}$$converges", "input": "\\sum_{n=0}^{\\infty\\:}ne^{-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": "$$\\left|\\frac{a_{n+1}}{a_n}\\right|=\\left|\\frac{\\left(n+1\\right)e^{-\\left(n+1\\right)}}{ne^{-n}}\\right|$$" }, { "type": "interim", "title": "Simplify $$\\left|\\frac{\\left(n+1\\right)e^{-\\left(n+1\\right)}}{ne^{-n}}\\right|:{\\quad}\\frac{\\left|n+1\\right|}{e\\left|n\\right|}$$", "steps": [ { "type": "step", "result": "=\\left|\\frac{\\left(n+1\\right)e^{-\\left(n+1\\right)}}{ne^{-n}}\\right|" }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=\\frac{1}{x^{b-a}}$$", "secondary": [ "$$\\frac{e^{-\\left(n+1\\right)}}{e^{-n}}=\\frac{1}{e^{-n-\\left(-\\left(n+1\\right)\\right)}}$$" ], "result": "=\\frac{n+1}{ne^{-n-\\left(-\\left(n+1\\right)\\right)}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add similar elements: $$-n-\\left(-\\left(n+1\\right)\\right)=1$$", "result": "=\\left|\\frac{n+1}{en}\\right|" }, { "type": "step", "primary": "Apply absolute rule: $$\\left|\\frac{a}{b}\\right|\\:=\\frac{\\left|a\\right|}{\\left|b\\right|}$$", "result": "=\\frac{\\left|n+1\\right|}{\\left|en\\right|}" }, { "type": "step", "primary": "Apply absolute rule: $$\\left|a\\:\\cdot\\:b\\right|\\:=\\left|a\\right|\\:\\left|b\\right|$$", "secondary": [ "$$\\left|en\\right|=\\left|e\\right|\\left|n\\right|$$" ], "result": "=\\frac{\\left|n+1\\right|}{\\left|e\\right|\\left|n\\right|}" }, { "type": "step", "primary": "Apply absolute rule: $$\\left|a\\right|=a,\\:a\\ge0$$", "secondary": [ "$$\\left|e\\right|=e$$" ], "result": "=\\frac{\\left|n+1\\right|}{e\\left|n\\right|}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{\\left|n+1\\right|}{e\\left|n\\right|}\\right)=\\frac{1}{e}$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{\\left|n+1\\right|}{e\\left|n\\right|}\\right)", "steps": [ { "type": "step", "primary": "$$n+1$$ is positive when $$n\\to\\:\\infty\\:$$. Therefore $$\\left|n+1\\right|=n+1$$", "result": "=\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{n+1}{e\\left|n\\right|}\\right)" }, { "type": "step", "primary": "$$n$$ is positive when $$n\\to\\:\\infty\\:$$. Therefore $$\\left|n\\right|=n$$", "result": "=\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{n+1}{en}\\right)" }, { "type": "step", "primary": "$$\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$", "result": "=\\frac{1}{e}\\cdot\\:\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{n+1}{n}\\right)" }, { "type": "interim", "title": "Divide by highest denominator power: $$1+\\frac{1}{n}$$", "input": "\\frac{n+1}{n}", "result": "=\\frac{1}{e}\\cdot\\:\\lim_{n\\to\\:\\infty\\:}\\left(1+\\frac{1}{n}\\right)", "steps": [ { "type": "step", "primary": "Divide by $$n$$", "result": "=\\frac{\\frac{n}{n}+\\frac{1}{n}}{\\frac{n}{n}}" }, { "type": "step", "primary": "Refine", "result": "=1+\\frac{1}{n}" } ], "meta": { "interimType": "Divide By Highest Denominator Power 1Eq", "practiceLink": "/practice/limits-practice#area=main&subtopic=Rational%20functions", "practiceTopic": "Rational Functions Limits" } }, { "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": "=\\frac{1}{e}\\left(\\lim_{n\\to\\:\\infty\\:}\\left(1\\right)+\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{1}{n}\\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": "$$\\lim_{n\\to\\:\\infty\\:}\\left(1\\right)=1$$", "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": "$$\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{1}{n}\\right)=0$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{1}{n}\\right)", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$\\lim_{x\\to\\infty}\\left(\\frac{c}{x^a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jbUbsVfpMbXIJ0UHfO5sfBo3vAig8R1lz2YYpgaM3COyRKA+kVWL4EgNcz8UITyk3WldPTzMRCmfRYnoIUxcFAsqG8mG4+AKrheu2ZGyPikLL/bRq43OIhscddZMC8wf7LqB9CcyvYCWDsGseX09hi2Sg2N1jZXcumfy0+UpgDwaN7wIoPEdZc9mGKYGjNwHT+Ezdx8WMx3O68TWD7RsM2K6cO8RrGfyfucGJ9J7KQ=" } }, { "type": "step", "result": "=\\frac{1}{e}\\left(1+0\\right)" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{1}{e}", "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": "=\\mathrm{converges}" } ], "meta": { "solvingClass": "Series", "practiceLink": "/practice/series-practice#area=main&subtopic=Ratio%20Test", "practiceTopic": "Series Ratio Test" } } }