Finding whether the series $$\sum^{\infty}_{k=0}\frac{5k^2+7}{8k^2+2}$$ is converges or diverges.
What i Try:
I am Trying to solve it using ratio test
Let $\displaystyle a_{k}=\frac{5k^2+7}{8k^2+2}$. Then $\displaystyle a_{k+1}=\frac{5(k+1)^3+7}{8(k+1)^2+2}$.
Then $$\lim_{k\rightarrow \infty}\bigg|\frac{a_{k+1}}{a_{k}}\bigg|=1$$
But this test does not gave any conclusion.
Please help me How do i solve it. Thanks
$\frac{5k^2+7}{8k^2+2}\ge\frac{5k^2+5}{8k^2+8}=\frac{5}{8}$
By comparision test, since $\sum_{k=0}^{k=\infty}5/8$ diverges hence
$\sum^{\infty}_{k=0}\frac{5k^2+7}{8k^2+2}$ also diverges.
Alternatively, simply note that $n$th term does not tend to zero for this series as $n\to \infty$, which is necessary (not sufficient!) condition for convergence of a series.
Recall that $$\sum_{i=0}^n \:a_n \text{ converges}\implies\lim_{n\to\infty}a_n=0$$ Since $$\lim_{n\to\infty}a_n=\lim_{n\to\infty}\frac{5n^2+7}{8n^2+2}=\frac{5}{8}\neq0$$ it follows that the infinite sum diverges (this is merely the contrapositive of the proposition above).
$$\frac{5k^2+7}{8k^2+2}=\frac 58-\frac{23}{32k^2+8}.$$
The first term causes divergence.