How to tell if a log series converges?

Question

I have the following series.

$$(-1)^n \times \ln\Bigg(\frac{8n+5}{7n+3}\Bigg)$$

I tried the root, ratio and integral tests, but am doing something wrong because I am unable to tell if this series converges.

Answer 1

Hint:

Use the nth term test.

First consider the inside fraction. What is $\displaystyle \lim_{n \rightarrow \infty} \Bigg( \frac{8n+5}{7n+3} \Bigg)$? Given this, what is the asymptotic behavior when the natural log is applied?

Answer 2

Considering a large value of $n$, rewrite $$ \ln\Bigg(\frac{8n+5}{7n+3}\Bigg)=\ln(8n+5)-\ln(7n+3)=\ln(8n)+\ln\Big(1+\frac{5}{8n}\Big)-\ln(7n)-\ln\Big(1+\frac{3}{7n}\Big)$$ $$ \ln\Bigg(\frac{8n+5}{7n+3}\Bigg)=\ln\Big(\frac{8}{7}\Big)+\ln\Big(1+\frac{5}{8n}\Big)-\ln\Big(1+\frac{3}{7n}\Big)$$ Now, consider, for small values of $y$, the Taylor series $$\ln(1+y) \simeq \frac{y}{1}-\frac{y^2}{2}+\frac{y^3}{3}$$ replace $y=\frac{5}{8n}$ in the first logarithm and $y=\frac{3}{7n}$ in the second logarithm. You will end with $$ \ln\Bigg(\frac{8n+5}{7n+3}\Bigg)= \log \left(\frac{8}{7}\right)+\frac{11}{56 n}-\frac{649}{6272 n^2}+\frac{29051}{526848 n^3}+O\left(\left(\frac{1}{n}\right)^4\right)$$

Applying the same approach to the most general case, you could find that $$ \ln\Bigg(\frac{an+b}{cn+d}\Bigg)= \log \left(\frac{a}{c}\right)+\frac{\frac{b}{a}-\frac{d}{c}}{n}+\frac{\frac{d^2}{2 c^2}-\frac{b^2}{2 a^2}}{n^2}+\frac{\frac{b^3}{3 a^3}-\frac{d^3}{3 c^3}}{n^3}+O\left(\left(\frac{1}{n}\right)^4\right)$$