dealing with trig series

[iDoof]

What kind of test should I use to determine the convergence of such infinite series as:

SUM of (sin(n*pi/2))/n! from n=1 to infinity?

It's an alternating series...but I'm not quite sure how to employ the Alternating Series Test because there's no explicit (-1)^n+1 power in it...does that make sense? If that DOESN'T make sense, could someone just walk me through how to do this?

Thanks so much

 

[Unco]

G'day, iDoof.

You see that it is an alternating series but you haven't quite completed the link.

Write out a few terms:
1 - 1/3! + 1/5! - 1/7! + ...

There's no trig there. Can you write a new summation for this?

 

[soroban]

Hello, iDoof!

Write out a few terms: .\(\displaystyle 1\,-\,\frac{1}{3!}\,+\,\frac{1}{5!}\,-\,\frac{1}{7!}\,+\,\cdots\)

Do you recognize this series?

Are you familiar with: .\(\displaystyle \sin x\:=\:x\,-\,\frac{x^3}{3!}\,+\,\frac{x^5}{5!}\,-\,\frac{x^7}{7!}\,+\,\cdots\)

That's right! . . . the sum is: \(\displaystyle \sin(1)\)

 

[Unco]

There's no trig here.

Apologies . . .

 

[iDoof]

ah i see thanks!