Determine the infinite sum of b_k = c_k - c_(k-1)

[lizarton]

Suppose that b_k = c_k - c_{k-1}, where \langle c \rangle is a sequence such that c_0=1 and \lim_{k → ∞} c_k = 0. Use the definition of series to determine \sum^{∞}_{k=1} b_k.

I've done a little analysis, so I think that c_k is decreasing, since the first term is greater than the limit of the sequence \Rightarrow b_k \leq 0 and decreasing (I think).

Also, by taking limit of both sides I obtain:
\lim_{k → ∞} b_k = \lim_{k → ∞} c_k - c_{k-1} = 0.
(I understand this does not conclude that the series converges)

From the looks of the problem, it smells like Cauchy convergence criterion, but I'm not sure where to start with such an abstract description of the sequences.

The hint says to compute the partial sums exactly, but since no explicit formula is given for \langle c \rangle, I'm not so sure how to go about this.

Here is the definition we use for an infinite series:

Let \langle a \rangle be a sequence of real numbers. The formal expression \sum^{∞}_{k=1} a_k is an infinite series. The number s_n = \sum^{n}_{k=1} a_k is the nth partial sum of the series. The infinite series \sum^{∞}_{k=1} a_k converges if \lim_{n → ∞} s_n exists; otherwise the series diverges. When \sum^{∞}_{k=1} a_k converges, we write L = \lim s_n=\sum^{∞}_{k=1} a_k and say that the sum of the series equals L.

Any help would be appreciated. Thanks so much!

*EDIT* Let me know if I'm onto something here:
Let s_n = \sum^{n}_{k=1} b_k and S_n = \sum^{n}_{k=0} c_k = 1 + \sum^{n}_{k=1} c_k.

Then S_{n-1} = \sum^{n-1}_{k=0} c_{k-1} = 1 + \sum^{n-1}_{k=1} c_k.

Then s_n = S_n - S_{n-1} = 1 + \sum^{n}_{k=1} c_k - 1 - \sum^{n-1}_{k=1} c_k = c_n.

\sum^{∞}_{k=1} b_k = \lim_{n→∞} (S_n - S_{n-1}) = \lim_{n→∞} c_n = 0

 

[micromass]

First of all, the c_n does not need to be decreasing.

Second, the hint is a good one:

b_0=c_0

b_1+b_0=(c_1-c_0)+c_0=c_1

b_2+b_1+b_0=(c_2-c_1)+(c_1-c_0)+c_0=c_2

Generalize.

EDIT: it seems you found it already...