Testing Absolute Convergence of ∑(-2)n+1/n+5n

[Another]

∑ (-2)ⁿ⁺¹/n+5ⁿ Test this series Absolute Convergence ?

∑|a_n| = ∑(2)ⁿ⁺¹/n+5ⁿ

if the sum of |a_n| converges, than the sum of a_n converges

∑|a_n| = ∑(2)ⁿ⁺¹/n+5ⁿ

I can use Comparison Test?
I can choose series b_n = ∑ 2ⁿ/5ⁿ ?

 

[mfb]

Please don't delete the homework template.

Do you mean $\displaystyle \sum \frac{(-2)^{n+1}}{n+5^n}$? Then you need brackets around the denominator. If that is your series, you can use this comparison.

 

[Ray Vickson]

∑ (-2)ⁿ⁺¹/n+5ⁿ Test this series Absolute Convergence ?

∑|a_n| = ∑(2)ⁿ⁺¹/n+5ⁿ

if the sum of |a_n| converges, than the sum of a_n converges

∑|a_n| = ∑(2)ⁿ⁺¹/n+5ⁿ

I can use Comparison Test?
I can choose series b_n = ∑ 2ⁿ/5ⁿ ?

What you wrote is obviously divergent: your series has
$$a_n = \frac{(-2)^{n+1}}{n} + 5^n,$$
giving two divergent series.

Perhaps you meant
$$a_n = \frac{(-2)^{n+1}}{n+5^n},$$
but that is not what you wrote. You need parentheses: just write (-2)^(n+1)/(n+5^n), and the problem would go away!

 

[Another]

Thank you very much

Ray Vickson
and
mfb