Series convergence/ divergence $\sum_{n=1}^\infty(-0.75)^n\frac{n+1}2$

Question

Determine whether the following series is absolutely convergent, conditionally convergent or divergent. $$\sum_{n=1}^\infty(-0.75)^n\dfrac{n+1}{2}$$

I tried using the divergence test and the alternating series test but I can't seem to use it or this problem as I could not take the limit of the function.

Answer 1

Let $$u_n=(-0.75)^n\frac{n+1}{2}$$ then by the ratio test we have easily: $$\lim_{n\to\infty}\left|\frac{u_{n+1}}{u_n}\right|=0.75<1$$ hence the given series is absolutely convergent.

Answer 2

Your series is $$\frac{1}{2}\sum_{n=1}^\infty(-1)^n(3/4)^n(n+1)=\frac{1}{2}\underbrace{\sum_{n=1}^\infty(-1)^n(3/4)^nn}_{T_1}+\frac{1}{2}\underbrace{\sum_{n=1}^\infty(-1)^n(3/4)^n}_{T_2}$$ Obviously $T_2$ is absolutely convergent. And $T_1$ is also absolutely convergent since $$\sum_{n=1}^{\infty}nx^{n}=\frac{x}{(1-x)^2} \ \ \text{and} \ \ \ \sum_{n=0}^{\infty}x^n=\frac{1}{1-x}\ \ \text{where}\ \ \ |x|<1$$