Question:-1(d)

We are tasked with finding the range of $p(>0)$$p(>0)$p( > 0)p(>0)$p(>0)$ for which the series given by:

converges absolutely and conditionally. To do this, we will use various convergence tests and mathematical techniques.

Let $\sum u_n$$\sum u_n$sumu_(n)\sum u_n$\sum u_n$ be the given series, and define a new series $v_n$$v_n$v_(n)v_n$v_n$ as follows:

Additionally, define ${\omega }_n$${\omega }_n$omega _(n)\omega_n${\omega }_n$ as:

By comparing $v_n$$v_n$v_(n)v_n$v_n$ with ${\omega }_n$${\omega }_n$omega _(n)\omega_n${\omega }_n$, we find that:

Using the comparison test, we conclude that $\sum v_n$$\sum v_n$sumv_(n)\sum v_n$\sum v_n$ is convergent when $p>1$$p>1$p > 1p>1$p>1$.

$\sum u_n$$\sum u_n$sumu_(n)\sum u_n$\sum u_n$ is an alternating series and $\sum \left|u_n\right|$$\sum \left|u_n\right|$sum|u_(n)|\sum\left|u_n\right|$\sum \left|u_n\right|$ is convergent. Therefore, $\sum u_n$$\sum u_n$sumu_(n)\sum u_n$\sum u_n$ is absolutely convergent.

When $0<p\le 1$$0<p\le 1$0 < p <= 10<p \leq 1$0<p\le 1$, the sequence $\{v_n\}$$\{v_n\}${v_(n)}\{v_n\}$\{v_n\}$ is a monotone decreasing sequence of positive real numbers with $lim{v}_n=0$$lim{v}_n=0$limv_(n)=0\lim v_n = 0$lim{v}_n=0$. By Leibniz’s test, $\sum (-1{)}^{n+1}{v}_n$$\sum (-1{)}^{n+1}{v}_n$sum(-1)^(n+1)v_(n)\sum (-1)^{n+1} v_n$\sum (-1{)}^{n+1}{v}_n$, which is equivalent to $\sum u_n$$\sum u_n$sumu_(n)\sum u_n$\sum u_n$, is convergent.

Since $\sum \left|u_n\right|$$\sum \left|u_n\right|$sum|u_(n)|\sum\left|u_n\right|$\sum \left|u_n\right|$ is divergent in this case, $\sum u_n$$\sum u_n$sumu_(n)\sum u_n$\sum u_n$ is conditionally convergent.

Let $\mathrm{\Sigma }{u}_n$$\mathrm{\Sigma }{u}_n$Sigmau_(n)\Sigma u_n$\mathrm{\Sigma }{u}_n$ be a series of positive real numbers. We can decompose it into two new series:

In summary, the given series $\sum u_n$$\sum u_n$sumu_(n)\sum u_n$\sum u_n$ is absolutely convergent when $P>1$$P>1$P > 1P > 1$P>1$, and it is conditionally convergent when $0<p\le 1$$0<p\le 1$0 < p <= 10<p \leq 1$0<p\le 1$. This analysis provides a comprehensive understanding of the convergence behavior of the series for different values of $p$$p$pp$p$ in the specified range.