Q. 6 - Calculus

Question

Explain why the sequence of partial sums of the series $\sum _{k=1}^{\infty } a_k$ in which $a_k>0$ for every $k\in {\mathrm{\mathbb{Z}}}^{+}$ is strictly increasing.

Step-by-Step Solution

Answer

The sequence of partial sums of a series $\sum _{k=1}^{\infty } a_k$s the partial sum of the first through nth terms.

The sequence of partial sums is given by, $\left\{S_n\right\}=\left\{a_1,\left(a_1+a_2\right),...,\left(a_1+a_2+...+a_n\right)\right\}$. As each $a_k>0$.

The sequence of partial sums of the given series is strictly increasing.

Step 1. Given information.

Consider the given question,

The series $\sum _{k=1}^{\infty } a_k$ in which $a_k>0$.

Step 2. Explain the sequence of partial sums of the given series.

The sequence of partial sums of a series $\sum _{k=1}^{\infty } a_k$ is the partial sum of the first through nth terms.

It can be defined as, $S_n=a_1+a_2+...+a_n$.

The sequence of partial sums is given by,

$$\begin{gathered} \left\{S_n\right\}=\left\{\sum _{k=1}^n a_k\right\} \\ =\left\{a_1,\left(a_1+a_2\right),...,\left(a_1+a_2+...+a_n\right)\right\} \end{gathered}$$

As each $a_k>0$.

Therefore, the sequence of partial sums of the series $\sum _{k=1}^{\infty } a_k$ is strictly increasing.