# Sums of sequences

## 1. Properties of the notion of limit with addition

So far in this course you have seen several sequences that converge with proofs of their convergence. These include:

• The sequence =1 which converges to 0 as .
• The sequence = (for a fixed number 0 1 ) which also converges to 0 as .
• The constant sequence = which converges to as .

We want to build up our repertoire of such sequences, and we start by looking at addition.

Theorem.

Let ( ) and ( ) be convergent sequences with limits and respectively. Let the sequence ( ) be defined by = + . Then + as .

Proof.

We are given that 0 - and 0 - and must prove that + + .

Subproof.

Let 0 be arbitrary.

Subproof.

Let 1 such that 1 - /2 and 2 such that 2 - /2 .

Let =( 1, 2) .

Subproof.

Let be arbitrary.

Subproof.

Assume .

Then 1 and 2 so - /2 and - /2 .

So ( + )-(+) - + - /2+/2= by the triangle inequality.

It follows that + -(+) .

So 0 + -(+) , as required.

This result enables us to write down the limit of sequences such as 1+1 and + . It also enables us to compute the limit of 2 writing this as + , and 3 , 4 , etc. The following result (which will be improved in a later web page) roll all these together in a single proposition.

Proposition.

Let ( ) be a convergent sequence with limit , and let be a number. Let the sequence ( ) be defined by = . Then as .

Proof.

We are given that 0 - and = for all . We will assume 0 , for if =0 then is the constant sequence 0 which converges to 0= .

Subproof.

Let 0 be arbitrary.

Subproof.

Let such that - / since 0 hence / 0 .

Then for each , - = - , i.e., - .

So - .

So 0 - , as required.

These two results show a similar theorem also holds for subtraction.

Theorem.

Let ( ) and ( ) be convergent sequences with limits and respectively. Let the sequence ( ) be defined by = - . Then - as .

Proof.

= - = +(-1) , so (-1) - by the proposition on multiplying -1 by the sequence , and = +(-1) +(-)=- , by the theorem on addition.