# Thread: Ratio and Root Test series problem.

1. ## 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.

2. 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

3. ## suggestion

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.

4. 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.
$\sum_{n=0}^{\infty} {\frac{1}{2^{4n+1}} + \frac{1}{3^{2n}}}$

5. Originally Posted by lilaziz1
$\sum_{n=0}^{\infty} {\frac{1}{2^{4n+1}} + \frac{1}{3^{2n}}}$
For the term where n=0, this will add an extra term equal to 1, so I'm not sure that works...

6. 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 ..

7. $\sum_{n=0}^{\infty} {\frac{1}{2^{4n+1}} + \frac{1}{3^{2n}}}$

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.