Infinite series- converges or not
I have 2 questions I am having problems with.
The goal is to determine if the series converges or not.
Q1: Sum from(1 to inf) of (exp^i)/( (exp^2i) + 9)
I tried to do the integral test but I cannot seem to integrate. Any guides would be appreciated.
Also if I wanted to compare it what would I compare it to.
Q2: Sum from (2 to inf) of 1/( (i^2) - 1)
I did the integral test and came out to the following
integral of 1/i^2 -1 = 1/2 (ln(i-1) - ln(i+1))
So in looking at this as i approaches infinity, can I say that integral approaches 0 and therefore this series converges.
All help is appreciated.
There's an easy comparison for both of them. Q1) compare to exp(i)/exp(2i), Q2) compare to 2/(i^2).
Dick: thanks for response. I will look at your way. Always good to learn new ways
Tom: This is how I did the integral test
Let u = exp^(2x) + 9
du/dx = 2exp^(2x)
This is where I got stuck. I thought once I do du/dx, I should get e^x in this equation but since I didn't I couldn't get anywhere.
So I tried by integration by parts and went into an infinite loop
a- let u = 1/( exp(2x) + 9)
b- du/dx = -2/(exp(2x +9)
c- let dv = exp^x dx
d- v = e^x
uv - integral (v wrt dx)
integral (e^x/e^2x +9) = e^x/(e^2x + 9) +2 integral (e^x/e^2x +9)
This is where I got stuck.
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