Finding Discontinuities & Decreasing Intervals of a Sequence

[Bashyboy]

Homework Statement

\sum_{n=1}^{\infty} \frac{n^{k-1}}{n^k+c}, where k is a positive integer.

Homework Equations

The Attempt at a Solution

I found that it was discontinuous at x = (-c)^{1/k}; and to determine if the sequence is decreasing, I took the

derivative which is--I think--f'(x) = \frac{(k-1)x^{k-2}(x^k+c)-x^{k-1}(kx^{k-1}}{(x^k+c)^2}
I am not quite sure how to simplify this, nor am I certain on how to find the intervals which the sequence is decreasing.

 

[STEMucator]

Homework Statement

\sum_{n=1}^{\infty} \frac{n^{k-1}}{n^k+c}, where k is a positive integer.

Homework Equations

The Attempt at a Solution

I found that it was discontinuous at x = (-c)^{1/k}; and to determine if the sequence is decreasing, I took the

derivative which is--I think--f'(x) = \frac{(k-1)x^{k-2}(x^k+c)-x^{k-1}(kx^{k-1}}{(x^k+c)^2}
I am not quite sure how to simplify this, nor am I certain on how to find the intervals which the sequence is decreasing.

Now that you've taken the derivative of f, ask yourself, what are the critical points? Those will allow you to find if f is decreasing as x → ∞.