Question

Log3+log4/3+log5/3+........infinity check whether its convergent or divergent

Answer 1

The series $\log 3+\log(\frac{4}{3})+\log(\frac{5}{3})+.....+\infty$ is divergent as the general term of the series will approach infinity for each term.

To determine if the series $\log 3+\log(\frac{4}{3})+\log(\frac{5}{3})+.....+\infty$ converges or diverges, we can examine the behaviour of the individual terms and apply the convergence tests.

If the partial sums of the series eventually stabilize and do not increase without bound, then the series is said to converge. On the other hand, if the partial sums grow without limit, the series is said to diverge.

The general term of the series is: $\log\frac{n+3}{n}$

First, let's rewrite the series in a more compact form:

$\log 3+\log(\frac{4}{3})+\log(\frac{5}{3})+...+\log\frac{n+3}{n}+...+\infty$

Now, let's investigate the behaviour of the terms.

As n increases, the value of  $\frac{n+3}{n}$ also increases. Therefore, the individual terms $\log\frac{n+3}{n}$  will approach positive infinity.

Since the terms of the series are not approaching zero, the series does not satisfy the necessary condition for convergence. Therefore, we can conclude that the series:

$\log 3+\log(\frac{4}{3})+\log(\frac{5}{3})+.....+\infty$ is divergent.

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