Is the Interval of Convergence for (x-2)^n / n^(3n) from -1 to 5?

[isukatphysics69]

Homework Statement

interval of convergence for
n=1 to inf
(x-2)ⁿ / n3ⁿ

Homework Equations

The Attempt at a Solution

i used the ratio test and solved for x and got that the interval of convergence is from -1 to 5. now i have to test the endpoints to determine which ones will make the series either converge or diverge. it looks to me that they both will diverge but on the multiple choice homework i do not see a choice for both diverging although i do see a choice for the -1 converging and the 5 diverging. now the 5 endpoint makes sense to me for divergence because it will be 1/n which is a divergent harmonic series
but the -1 converging is not making sense

(-3)ⁿ / n3ⁿ
(-1)¹ / n
divergent

what am i not understanding here?

 

[Ray Vickson]

Homework Statement

interval of convergence for
n=1 to inf
(x-2)ⁿ / n3ⁿ

Homework Equations

The Attempt at a Solution

i used the ratio test and solved for x and got that the interval of convergence is from -1 to 5. now i have to test the endpoints to determine which ones will make the series either converge or diverge. it looks to me that they both will diverge but on the multiple choice homework i do not see a choice for both diverging although i do see a choice for the -1 converging and the 5 diverging. now the 5 endpoint makes sense to me for divergence because it will be 1/n which is a divergent harmonic series
but the -1 converging is not making sense

(-3)ⁿ / n3ⁿ
(-1)¹ / n
divergent

what am i not understanding here?

You are not understand that the series $\sum 1/n$ diverges but the series $\sum (-1)^n/n$ converges. (Recall the "alternating series" test.)

In fact, the second sum is well-known: $\sum_{n=1}^{\infty} (-1)^n/n = - \ln 2.$

 

[haushofer]

You don’t understand the difference between conditional and absolute convergence.