Does the given series converge absolutely or conditionally?

[whatlifeforme]

Homework Statement

Determine either absolute convergence, conditional convergence or divergence for the series.

Homework Equations

\displaystyle \sum^{∞}_{n=1} \frac{(-1)^n}{5n^{1/4} + 5}

The Attempt at a Solution

It converges conditionally i know, but i can't figure out how.
1. I applied the alternating series test and concluded that the series converges.
2. I attempted to use the Ratio test for absolute convergence, but l'hospital's is just going in circles. I'm not getting anywhere; the limit n->∞ is ∞/∞.

 

[Dick]

Homework Statement

Determine either absolute convergence, conditional convergence or divergence for the series.

Homework Equations

\displaystyle \sum^{∞}_{n=1} \frac{(-1)^n}{5n^(1/4) + 5}

The Attempt at a Solution

It converges conditionally i know, but i can't figure out how.
1. I applied the alternating series test and concluded that the series converges.
2. I attempted to use the Ratio test for absolute convergence, but l'hospital's is just going in circles. I'm not getting anywhere; the limit n->∞ is ∞/∞.

If you mean (-1)^n/(5n^(1/4)+5) try a comparison test with a p-series. The ratio test won't help you.

 

[Mark44]

Homework Statement

Determine either absolute convergence, conditional convergence or divergence for the series.

Homework Equations

\displaystyle \sum^{∞}_{n=1} \frac{(-1)^n}{5n^(1/4) + 5}

Is this what you meant?
$$\sum^{∞}_{n=1} \frac{(-1)^n}{5n^{1/4} + 5} $$

The Attempt at a Solution

It converges conditionally i know, but i can't figure out how.
1. I applied the alternating series test and concluded that the series converges.
2. I attempted to use the Ratio test for absolute convergence, but l'hospital's is just going in circles. I'm not getting anywhere; the limit n->∞ is ∞/∞.

 

[whatlifeforme]

Is this what you meant?
$$\sum^{∞}_{n=1} \frac{(-1)^n}{5n^{1/4} + 5} $$

thanks. i fixed it.

 

[whatlifeforme]

If you mean (-1)^n/(5n^(1/4)+5) try a comparison test with a p-series. The ratio test won't help you.

how do i prove the p-series? use an integral test? i don't think my instructor will let us simply identify a p-series but wants us to rather prove it.

 

[Mark44]

how do i prove the p-series? use an integral test? i don't think my instructor will let us simply identify a p-series but wants us to rather prove it.

A p-series (below) converges or diverges, depending on the value of p.
$$ \sum_{n = 1}^{\infty} \frac{1}{n^p}$$
Use the integral test, with f(x) = x^-p, and look at cases for p < 1, p = 1, and p > 1. There might already be a proof of this in your book.