Test for convergence of the series

[DarkStalker]

Homework Statement

Q)
Summation from 1 to infinity
(1+(-1)^i) / (8i+2^i)

This series apparently converges and I can't figure out why.

Homework Equations

The Attempt at a Solution

(1+(-1)^i) / i(8+2^i/i)

Taking the absolute value of the above generalization:

2/i(8+2^i/i)

Rearranging that would give:

2/i * (1/(8+2^i/i)

Now I thought that since 2/i would diverge, the entire series should diverge.

Comparing it to the geometric series (1/2^i) implies converges, but I don't know what's wrong with the above

 

[vela]

2/i * (1/(8+2^i/i)

Now I thought that since 2/i would diverge, the entire series should diverge.

I don't understand your reasoning here. It's like saying
$$\sum_{n=1}^\infty \frac{1}{2^n}$$ diverges because you can write it as
$$\sum_{n=1}^\infty \left(\frac{1}{2}\times\frac{1}{2^{n-1}}\right)$$ and $$\sum_{n=1}^\infty \frac{1}{2}$$ diverges.

 

[DarkStalker]

I don't understand your reasoning here. It's like saying
$$\sum_{n=1}^\infty \frac{1}{2^n}$$ diverges because you can write it as
$$\sum_{n=1}^\infty \left(\frac{1}{2}\times\frac{1}{2^{n-1}}\right)$$ and $$\sum_{n=1}^\infty \frac{1}{2}$$ diverges.

Point. I just checked my textbook and it appears I'd mistakenly thought that just because Summation 1-infinity (a_i+b_i) equals summation 1-infinity (a_i) + Summation 1-infinity (b_i), I thought the same would be true for multiplication.

Thanks for clearing it up.