# Ratio and Root Test series problem.

Printable View

• Apr 7th 2010, 11:31 PM
Tulki
Ratio and Root Test series problem.
Here it is:

Show that the ratio test is inconclusive but the root test indicates convergence for the series:

(1/2) + 1/(3^2) + 1/(3^4) + 1/(2^5) + . . .

The problem I'm having here isn't necessarily the application of the ratio and root tests. Rather, I'm having trouble recognizing any sort of pattern in this series so I can get started! It should be pretty simple if I can somehow acquire an expression for the "nth" term in the series...

Any and all help is greatly appreciated.
• Apr 8th 2010, 02:12 AM
tonio
Quote:

Originally Posted by Tulki
Here it is:

Show that the ratio test is inconclusive but the root test indicates convergence for the series:

(1/2) + 1/(3^2) + 1/(3^4) + 1/(2^5) + . . .

The problem I'm having here isn't necessarily the application of the ratio and root tests. Rather, I'm having trouble recognizing any sort of pattern in this series so I can get started! It should be pretty simple if I can somehow acquire an expression for the "nth" term in the series...

Any and all help is greatly appreciated.

As far as I can see there is no recognizable pattern in those terms (it can be several things), so the question is impossible to solve.

Tonio
• Apr 8th 2010, 04:11 AM
Pulock2009
suggestion
Quote:

Originally Posted by Tulki
Here it is:

Show that the ratio test is inconclusive but the root test indicates convergence for the series:

(1/2) + 1/(3^2) + 1/(3^4) + 1/(2^5) + . . .

The problem I'm having here isn't necessarily the application of the ratio and root tests. Rather, I'm having trouble recognizing any sort of pattern in this series so I can get started! It should be pretty simple if I can somehow acquire an expression for the "nth" term in the series...

Any and all help is greatly appreciated.

can something be done by rearranging the terms???say by grouping the powers of 2 together and those of 3 together then seperately doing something???Like (1/2+/(2^5)+....)+(1/(3^2)+1/(3^4)....).then for the series involving 2 we see that 1/(2^(2n+1)) where n begins from 0 could be a possible pattern. and for the series involving 3 we see that 1/(3^(2n)) could be a possible pattern.
• Apr 8th 2010, 06:43 AM
lilaziz1
Quote:

Originally Posted by Tulki
Here it is:

Show that the ratio test is inconclusive but the root test indicates convergence for the series:

(1/2) + 1/(3^2) + 1/(3^4) + 1/(2^5) + . . .

The problem I'm having here isn't necessarily the application of the ratio and root tests. Rather, I'm having trouble recognizing any sort of pattern in this series so I can get started! It should be pretty simple if I can somehow acquire an expression for the "nth" term in the series...

Any and all help is greatly appreciated.

$\displaystyle \sum_{n=0}^{\infty} {\frac{1}{2^{4n+1}} + \frac{1}{3^{2n}}}$
• Apr 8th 2010, 12:53 PM
Tulki
Quote:

Originally Posted by lilaziz1
$\displaystyle \sum_{n=0}^{\infty} {\frac{1}{2^{4n+1}} + \frac{1}{3^{2n}}}$

For the term where n=0, this will add an extra term equal to 1, so I'm not sure that works...
• Apr 8th 2010, 01:18 PM
General
Quote:

Originally Posted by Tulki
For the term where n=0, this will add an extra term equal to 1, so I'm not sure that works...

The finite sum does not affect the convergence of the series ..
• Apr 8th 2010, 01:54 PM
Tulki
$\displaystyle \sum_{n=0}^{\infty} {\frac{1}{2^{4n+1}} + \frac{1}{3^{2n}}}$

I found that the ratio test was inconclusive quite easily, but now I'm having trouble getting the "nth" term into the form (an)^n in order to apply the root test. I can get each of the individual sum terms into that form, but it doesn't seem as if that helps get me anywhere.