Prove an infinite sum exists and its sum

[B3NR4Y]

Homework Statement

Let {b _k } be a sequence of positive numbers. Assume that there exists a sequence {a _k}, such that a _k is greater than or equal to 0 for all k, a_k is decreasing, the limit of a_k is 0 and b_k = a_k - a _(k+1). Show that the sum from k=1 to infinity of b _k exists and equals a_1

Homework Equations

Not really any I can think of

The Attempt at a Solution

I'm not sure how to prove this. The sum of b_k has two parts that both go to zero, but I can think of an a_k that satisfies all those properties but doesn't converge (1/k), but if you write out the sum you can clearly see all the terms cancel except the a_1 term.

For example:

Sum of b_k = (a₁ - a ₂ ) + (a_2-a_3)+(a_3-a_4)+...

Which clearly cancels all the terms except a_1. But this doesn't seem rigoruous enough.(Also sorry for inconsistent subscripts, I'm on my phone)

 

[RUber]

I would start by doing just what you did. Compute the partial sum,
$\sum_{i=1}^{N-1} b_k$
Then show that in the limit as N goes to infinity, you get what you expect.

 

[Ray Vickson]

Homework Statement

Let {b _k } be a sequence of positive numbers. Assume that there exists a sequence {a _k}, such that a _k is greater than or equal to 0 for all k, a_k is decreasing, the limit of a_k is 0 and b_k = a_k - a _(k+1). Show that the sum from k=1 to infinity of b _k exists and equals a_1

Homework Equations

Not really any I can think of

The Attempt at a Solution

I'm not sure how to prove this. The sum of b_k has two parts that both go to zero, but I can think of an a_k that satisfies all those properties but doesn't converge (1/k), but if you write out the sum you can clearly see all the terms cancel except the a_1 term.

For example:

Sum of b_k = (a₁ - a ₂ ) + (a_2-a_3)+(a_3-a_4)+...

Which clearly cancels all the terms except a_1. But this doesn't seem rigoruous enough.(Also sorry for inconsistent subscripts, I'm on my phone)

Don't forget that the infinite sum $\sum_{i=1}^{\infty} b_i$ is defined as the limit $\lim_{n \to \infty} \sum_{i=1}^n b_i$. That is, the infinite sum exists if and only if the limit of finite sums exist.