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February 19th, 2013, 12:22 PM   #1
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series convergence?

Hi i am trying to figure out how this series would behave:

given that




can anyone give me a hand?
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February 19th, 2013, 12:25 PM   #2
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Re: series convergence?

Ignoring the first few terms , doesn't that seem like the derivative of a geometric series ?
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February 19th, 2013, 12:41 PM   #3
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Re: series convergence?

kind but I dont think changing the index will help because of the 'n' I think the geometric series formulas dont apply
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February 19th, 2013, 10:20 PM   #4
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Re: series convergence?

You want to know the convergence, right?



Now, since both sin and cos are bounded by 1, the imaginary, and the real part can be bounded by the sum .

Since we are given that is smaller than 1, the sum above converges by ratio test. Hence, the original series converges.

You said you are interested in the behavior, do you meant that you want to know the growth?

Challenge : Find an analytic formula for this series!
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February 20th, 2013, 03:29 AM   #5
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Re: series convergence?

By comparison test :



The sum is absolutely convergent since the latter is a derivative of a geometric series ...
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