Infinite Series with Numerator Arithmetic and Denominator Geometric

This series is supposed to be an infinite series but it's not going up by a set common ratio. How do I find the sum of the infinite series of this series?

Re: Infinite Series with Numerator Arithmetic and Denominator Geometric

Hello, Espionage!

The geometric series has first term and common ratio

Its sum is: .

Hence: .

Therefore: .

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Re: Infinite Series with Numerator Arithmetic and Denominator Geometric

Edit: Post above answered it way better.

Re: Infinite Series with Numerator Arithmetic and Denominator Geometric

If I was going to find the infinite sum, I would begin with the difference equation to obtain the partial sum in closed form:

where

The corresponding homogeneous solution is:

We then look for a particular solution of the form:

We may now determine the coefficients A and B.

Substituting this into the original difference equation, we find:

Multiply through by :

Equating coefficients, we obtain the system:

And thus:

By superposition, we have:

Using the initial value, we find:

and so we find:

Hence, the infinite sum is:

Re: Infinite Series with Numerator Arithmetic and Denominator Geometric