1 THINKING BACK - Calculus

Question

A strictly increasing sequence of partial sums: if \[ a_k > 0\] for \[k \;\epsilon\; \mathbb{Z}^{+}\]

Explain why the sequence of partial sums for the series \[\sum_{k=1}^{\infty}a_{k}\] is strictly increasing.

Step-by-Step Solution

Answer

The sequence of partial sum is strictly increasing.

Step 1. Given Information

The term a_k is always greater than 0 and the given sequence is

 \[\sum_{k=1}^{\infty}a_{k}\] 

Step 2. Proof

To prove that the sequence of partial sums of the series 

 \[\sum_{k=1}^{\infty}a_{k}\]  is strictly increasing, we must show that the nth partial sum is strictly larger than the (n-1)th partial sum for all n ∈ Z+. 

Let Sn be the nth partial sum, that is, 

\[S_{n}= a_{1} + a_{2} +..... + a_{n}\]

To prove that Sn is strictly larger than Sn-1, we need to show that Sn - Sn-1 > 0.

Subtracting Sn-1 from both sides, we get

\[S_{n}-S_{n-1}= (a_{1} + a_{2} +..... + a_{n} )-(a_{1} + a_{2} +.... + a_{n-1})\]

\[= a_{n}-0\]

\[=a_{n}\]

Since we are given that , a_k > 0 for all k ∈ Z⁺, it follows that a_n > 0 for all n ∈ Z⁺. Therefore, S_n - S_n-1 > 0, which means that the sequence of partial sums is strictly increasing. 

Hence Proved.