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Question Number 142629 by qaz last updated on 03/Jun/21

\underset{n = 1}{\sum^{\infty}} {(- 1)}^{n} \cdot \frac{2 n - 1}{(2 n)!} \cdot {(\frac{π}{2})}^{2 n}

= \underset{n = 1}{\sum^{\infty}} \frac{2 n - 1}{(2 n)!} \cdot {(- {(\frac{π}{2})}^{2})}^{n}

= (2 xD - 1) ∣_{x = π / 2} \underset{n = 1}{\sum^{\infty}} \frac{{(- x^{2})}^{n}}{(2 n)!}

= (2 xD - 1) ∣_{x = π / 2} [\underset{n = 0}{\sum^{\infty}} \frac{{(- x^{2})}^{n}}{(2 n)!} - 1]

= (2 xD - 1) ∣_{x = π / 2} (\cos x - 1)

= (- 2 xsin x - \cos x + 1) ∣_{x = π / 2}

= 1 - π

where is wrong ?

Answered by Dwaipayan Shikari last updated on 03/Jun/21

\underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n} (2 n)}{(2 n)!} {(\frac{π}{2})}^{2 n} - \frac{{(- 1)}^{n}}{(2 n)!} {(\frac{π}{2})}^{2 n}

= - (\frac{2}{2!} {(\frac{π}{2})}^{2} - \frac{4}{4!} {(\frac{π}{2})}^{4} + \frac{6}{6!} {(\frac{π}{2})}^{6} + \dots) + (\frac{1}{2!} {(\frac{π}{2})}^{2} - \frac{1}{4!} {(\frac{π}{2})}^{4} + . .)

= - \frac{π}{2} (\frac{π}{2} - \frac{1}{3!} {(\frac{π}{2})}^{3} + . .) - (1 - \frac{1}{2!} {(\frac{π}{2})}^{2} - \dots) + 1

= - \frac{π}{2} s i n (\frac{π}{2}) - c o s (\frac{π}{2}) + 1 = 1 - \frac{π}{2}

Commented by qaz last updated on 03/Jun/21

i have another solution .

\underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n} (2 n - 1)}{(2 n)!} x^{2 n} ∣_{x = \frac{π}{2}}

= \underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n + 1} (2 n + 1)}{(2 n + 2)!} x^{2 n + 2} ∣_{x = \frac{π}{2}}

= - x^{2} \underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n}}{(2 n + 2) (2 n)!} x^{2 n}

= - x^{2} \int_{0}^{1} y \underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n}}{(2 n)!} \cdot {(xy)}^{2 n} dy

= - x^{2} \int_{0}^{1} ycos (xy) dy

= - x^{2} {{[\frac{1}{x} ysin (xy) + \frac{1}{x^{2}} \cos (xy)]}_{0}^{1}}_{x = \frac{π}{2}}

= - x^{2} {\frac{1}{x} \sin x + \frac{1}{x^{2}} (\cos x - 1)}_{x = \frac{π}{2}}

= 1 - \frac{π}{2}

- - - - - - - - - - - - - - - - - - -

but \underset{n = 1}{\sum^{\infty}} (2 n - 1) \frac{{(- 1)}^{n}}{(2 n)!} x^{2 n} = (xD - 1) (\cos x - 1),

i still dont understand, why not LFT = (2 xD - 1) (\cos x - 1) ? ? ?

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