Series Convergences

• Dec 13th 2008, 08:10 PM
Series Convergences
Its a normal summation for K=0 and goes to infinite for $[(-1)^k]/(2k+1)$

how do I show this converges... I know it does but I forgot how, using limit comparison, comparison, divergence, ratio, root, abs. ratio, abs. root, or AST.

thanks again
• Dec 13th 2008, 08:20 PM
Mathstud28
Quote:

Originally Posted by ryukolink
Its a normal summation for K=0 and goes to infinite for $[(-1)^k]/(2k+1)$

how do I show this converges... I know it does but I forgot how, using limit comparison, comparison, divergence, ratio, root, abs. ratio, abs. root, or AST.

thanks again

Use the Alternatig Series Test using the fact that if $a_n=\frac{1}{2n+1}$ then $a_{n+1} and $\lim_{n\to\infty}a_n=0$

Note: $\sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1}=\frac{\pi}{4}$
• Dec 13th 2008, 08:31 PM
so Im digging through my backpack and I find a paper explaining the AST for oscillating harmonics lol

thanks though..

Now tell me this... is a multiple of a harmonic only when its 1/k multiplied by a number > 1 ?

so like 1/2k this would not be a multiple of a harmonic?

because by AST, we also take the lim as x approaches infinte... 1/2k+1 compares to 1/k .. 1/2k+1 < 1/k aww which proves nothing.. darn it but the limit comp. text proves its 1/2 . Thus the two act similiar so it diverges.. or did I mess that up somewhere?
• Dec 13th 2008, 08:53 PM
Mathstud28
Quote:

Originally Posted by ryukolink
so Im digging through my backpack and I find a paper explaining the AST for oscillating harmonics lol

That is always how it goes
• Dec 13th 2008, 08:55 PM
Mathstud28
Quote:

Originally Posted by ryukolink
so Im digging through my backpack and I find a paper explaining the AST for oscillating harmonics lol

thanks though.. you truly are a math stud.. :)(no homo)

Now tell me this... is a multiple of a harmonic only when its 1/k multiplied by a number > 1 ?

so like 1/2k this would not be a multiple of a harmonic?

because by AST, we also take the lim as x approaches infinte... 1/2k+1 compares to 1/k .. 1/2k+1 < 1/k aww which proves nothing.. dammit but the limit comp. text proves its 1/2 . Thus the two act similiar so it diverges.. or did I mess that up somewhere?

No if you have $\sum \frac{a}{n}$ where $a$ is any real constant then your series converge. But I think you are confusing the AST, it states roughly: If you have a series $\sum (-1)^na_n$ and $a_{n+1}\leqslant a_{n}$ and $\lim a_n=0$ then the series converges.
• Dec 13th 2008, 08:58 PM
mr fantastic
Quote:

Originally Posted by ryukolink
so Im digging through my backpack and I find a paper explaining the AST for oscillating harmonics lol

thanks though..

Now tell me this... is a multiple of a harmonic only when its 1/k multiplied by a number > 1 ? Mr F says: This will not define an alternating series ....

so like 1/2k this would not be a multiple of a harmonic?

because by AST, we also take the lim as x approaches infinte... 1/2k+1 compares to 1/k .. 1/2k+1 < 1/k aww which proves nothing.. darn it but the limit comp. text proves its 1/2 . Thus the two act similiar so it diverges.. or did I mess that up somewhere?

..