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n-1-1-n-1-30-2n-1-2n-1-




Question Number 145451 by math55 last updated on 05/Jul/21
Σ_(n=1) ^∞ (−1)^(n−1) ((30^(2n−1) )/((2n−1)!))
$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \frac{\mathrm{30}^{\mathrm{2}{n}−\mathrm{1}} }{\left(\mathrm{2}{n}−\mathrm{1}\right)!} \\ $$
Answered by Olaf_Thorendsen last updated on 05/Jul/21
S = Σ_(n=1) ^∞ (−1)^(n−1) ((30^(2n−1) )/((2n−1)!))  S = Σ_(n=0) ^∞ (−1)^n ((30^(2n+1) )/((2n+1)!))  S = sin(30)  If is 30°, S = (1/2)
$$\mathrm{S}\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \frac{\mathrm{30}^{\mathrm{2}{n}−\mathrm{1}} }{\left(\mathrm{2}{n}−\mathrm{1}\right)!} \\ $$$$\mathrm{S}\:=\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}} \frac{\mathrm{30}^{\mathrm{2}{n}+\mathrm{1}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!} \\ $$$$\mathrm{S}\:=\:\mathrm{sin}\left(\mathrm{30}\right) \\ $$$$\mathrm{If}\:\mathrm{is}\:\mathrm{30}°,\:\mathrm{S}\:=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$
Commented by math55 last updated on 05/Jul/21
please sir is there anyway to prove that it's 1/2 without passing through sin30°

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