Q. 93 - Calculus

Question

 Prove that if ${{\left\{a_k\right\}}_{k=1}}^{\infty }$ is a sequence of positive real numbers, then the sequence ${{\left\{S_n\right\}}_{n=1}}^{\infty }$,  where the sequence Sn = a1 + a2 +···+an, is an increasing sequence.

Step-by-Step Solution

Answer

Proved.

Step 1. Given

Consider the sequences ${{\left\{a_k\right\}}_{k=1}}^{\infty }$.

Step 2. Proof

$$\begin{gathered} \mathrm{The} \mathrm{sequenc}e S_n =a_1 +a_2 +a_3 +...+a_n ......\left(1\right) \\ \mathrm{Change} \mathrm{n} \mathrm{to} \mathrm{n}+1 \mathrm{in} \mathrm{equation} \left(1\right) S_{n+1} =a_1 +a_2 +a_3 +...+a_n+a_{n+1} .....\left(2\right) \\ \mathrm{Subtract} \left(1\right) \mathrm{from} \left(2\right) \\ S_{n+1} -S_n =a_1 +a_2 +a_3 +...+a_n+a_{n+1}-(a_1 +a_2 +a_3 +...+a_n)=a. \\ \mathrm{Hence} S_{n+1} -S_n=a_{n+1} \\ \mathrm{Since} S_{n+1} \ge S_n \\ \mathrm{Hence} \mathrm{the} \mathrm{sequence} \mathrm{is} \mathrm{increasing} \mathrm{sequence} . \end{gathered}$$