Series Identities: Showing Convergence to cL & X+Y

[dtl42]

Homework Statement

a) If c is a number and $$\sum a_{n}$$ from n=1 to infinity is convergent to L, show that $$\sum ca_{n}$$ from n=1 to infinity is convergent to cL, using the precise definition of a sequence.

b)If $$\sum a_{n}$$ from n=1 to infinity and $$\sum b_{n}$$ from n=1 to infinity are convergent to X and Y respectively, show that $$\sum b_{n}+a_{n}$$ from n=1 to infinity is convergent to X+Y.

Homework Equations

I personally thought these were identities, and have no idea how to approach them.

The Attempt at a Solution

a) Maybe $$\sum a_{n}$$ from n=1 to infinity = $$Lim (S_{n})$$ as n goes to infinity, has something to do with it


I cross posted this in Calculus & Beyond and Pre-Calculus because I wasn't sure where it belongs

 

[HallsofIvy]

I bet the problem does NOT say "using the precise definition of a sequence". I'll bet it says "using the precise definition of convergence of a series". Write down the definition of convergence of a series as it applies to these two series. You are told that one converges and should be able to use that to show that the other converges.

"I personally thought these were identities, and have no idea how to approach them." I have no idea what you mean by that! Were you under the impression that one doesn't prove identities?

 

[dtl42]

The problem actually does say "Use the precise definition of limits for sequences in Sec. 10.2", that section covers Delta-Epsilon proofs of limits of sequences.