Find a formula for the nth partial sum of each series and use it to find the series’ sum if the series converges. 1−2+4−8+...+(−1)n−12n−1+...

Question

Find a formula for the nth partial sum of each series and use it to find the series’ sum if the series converges.  1−2+4−8+...+(−1)n−12n−1+...

i1ziZ · Accepted Answer

Given:The function given for the Taylor series is,f(x)=(1−x)−1a.The Taylor series for f(x)=(1−x)−1 centered at 0 is11−x=1+x+x2+x3+x4+...∑n=0∞xnNow, differentiate series with respect to x.ddx(1+x+x2+x3+x4+...)=1+2x+3x2+4x3+...=∑n=1∞nxn−1b.To find the function represented by the differentiated series, differentiate f(x).f′(x)=ddx(11−x)=1(1−x)2Therefore, the function represented by the differentiated series say h(x) ish(x)=1(1−x)2c. Find the interval of convergence of the power series for the derivative.The power series for the derivative isf′(x)=∑n=1∞nxn−1Use the Ratio test to compute the interval of convergence.=limn→∞|an+1||an|=limn→∞|(n+1)xn||nxn−1|=limn→∞(n+1n)|x|=limn→∞(n+(1+1n)n)|x|=limn→∞(1+1n)|x|=(1+0)|x|=|x|The  series converges for |x|&amp;lt;1.Therefore, -1

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Find a formula for the nth partial sum of each series and use it to find the series’ sum if the series converges. 1-2+4-8+...+(-1)^{n-1}2^{n-1}+...

Answered question

Answer & Explanation

New Questions in Precalculus