Show that $\displaystyle \int_{0}^{1} cos(x)dx$ converges.

A little bit puzzled here.

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- Apr 19th 2008, 06:12 PMlarsonImproper Integrals
Show that $\displaystyle \int_{0}^{1} cos(x)dx$ converges.

A little bit puzzled here. - Apr 19th 2008, 06:15 PMMathstud28
- Apr 19th 2008, 06:18 PMlarson
- Apr 19th 2008, 06:18 PMTheEmptySet
- Apr 19th 2008, 06:22 PMTheEmptySet
ahh the plot thickens

express cosine as its power series $\displaystyle \cos(x)=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n+1}}{(2n+1)!}$

so then

$\displaystyle \frac{\cos(x)}{x}=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n}}{(2n+1)!}$

integrate term by term and evaluate.

This will converges by the AST.

Good luck. (Rock) - Apr 19th 2008, 06:22 PMPaulRS
- Apr 19th 2008, 06:25 PMlarson
- Apr 19th 2008, 06:26 PMMathstud28
- Apr 19th 2008, 06:28 PMTheEmptySetMy above post is wrong I use the sine power series
express cosine as its power series $\displaystyle \cos(x)=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n}}{(2n)!}$

so then

$\displaystyle \frac{\cos(x)}{x}=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n-1}}{(2n)!}$

integrate term by term and evaluate and the sum diverges. - Apr 19th 2008, 06:29 PMTheEmptySet
- Apr 19th 2008, 06:30 PMlarson
- Apr 19th 2008, 06:32 PMMathstud28
- Apr 19th 2008, 06:34 PMlarson
- Apr 19th 2008, 06:35 PMPaulRS
- Apr 19th 2008, 06:39 PMMathstud28