\sum _{n=7}^{\infty }\:\frac{4}{11\sqrt{n}-5}
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MathBot Answer:
The infinite series \(\displaystyle\sum_{n=7}^{\infty} \frac{4}{11 \sqrt{n} - 5}\) diverges.
We can use Raabe-Duhamel's test to determine convergence of this series.
Raabe-Duhamel's test is an extension of the ratio test which can sometimes be used in cases where the ratio test is inconclusive.
Given a series \(\displaystyle\sum a_n\), define \(\rho = \displaystyle\lim_{n \to \infty} n\left(\frac{a_n}{a_{n+1}}-1\right)\)
•If \(\rho > 1\) the series converges.
•If \(\rho < 1\) the series diverges.
•If \(\rho = 1\) or the limit does not exist the test is inconclusive.
$$\begin{aligned}L &= \lim_{n \to \infty}\left|\frac{\frac{4}{11 \sqrt{n + 1} - 5}}{\frac{4}{11 \sqrt{n} - 5}}\right| \\ &= \lim_{n \to \infty} \left|{\frac{11 \sqrt{n} - 5}{11 \sqrt{n + 1} - 5}}\right| \\ &= \frac{1}{2}\end{aligned}$$Therefore the series diverges.