Prove the following series converges

Question

Let $u, v, w$ be real numbers such that $u+v+w=0$. Suppose that $\{b_k\colon k=0,1,2,\dots\}$ is a sequence of real numbers such that $\lim_{k\to\infty} b_k=0$. For $k=0,1,2,\dots$ define

$a_{3k}=u b_k,$ $a_{3k+1}= v b_k,$ $a_{3k+2} = w b_k.$

Prove that the series $ \sum_{n=0}^{\infty} a_n$ converges.

Im having a thouhg problem with this one, any idea I will appreciate.

So... this is one weird case of the alternating/Dirichlet test. — Simply Beautiful Art, Oct 26 '16 at 21:17
The most direct approach might be to compute explicitely the partial sums and to show that they converge to $0$ (yes, the series sums to $0$). — Did, Oct 26 '16 at 21:30
Just an idea: try to define different subsequences of the series and use squeeze theorem. — Masacroso, Oct 26 '16 at 21:32

score 4 · Answer 1 · answered Oct 26 '16 at 21:33

4

Let $S_n=\sum_{k=0}^n a_k.$

using $u+v+w=0$,

we get

$$S_{3n+2}=0,$$

$$S_{3n+1}=-wb_n$$

and

$$S_{3n}=ub_n.$$

thus, since $b_n \to 0,$ the three subsequences $(S_{3n}), (S_{3n+1})$ and $(S_{3n+2})$

converge to $0$, and we can say that

the partial sum sequence $(S_n)$

converge and the series $\sum a_n$ too.

answered Oct 26 '16 at 21:33

hamam_Abdallah

62,951

How did you use the $u+v+w=0$? – Thetexan Oct 26 '16 at 21:55
1

in $S_{3n+2}$ to get $0$. – hamam_Abdallah Oct 26 '16 at 21:57

score 0 · Answer 2 · answered Oct 26 '16 at 21:30

Let us have $c_{3k}=u,\ c_{3k+1}=v,\ c_{3k+2}=w$. By the Dirichlet test, we need to show that $\sum_{k=0}^N c_k$ is bounded and that $\lim_{n\to\infty}b_n=0$.

Clearly, we have

$$\sum_{k=0}^N c_k\le\sum_{k=0}^Mc_k$$

for some $M\in[0,1,2]$ and we already have $\lim_{n\to\infty}b_n=0$, meaning that

$$\sum_{k=0}^\infty c_kb_k$$

converges.

Prove the following series converges

2 Answers2