Tessellate S.T.E.M.S (2019) - Mathematics - Category C - Set 2 - Subjective Problem 2

Let $ \{a_n\}_{n \in \mathbb{N}}$ be a sequence of real numbers such that $b_m = \Sigma_{n = 1}^{m}a_n $ is a bounded sequence.

Let $\{c_k\}_{k \in \mathbb{N}}$ be another sequence of positive reals such that $\lim_{t \rightarrow \infty} c_t = 0$ .

Does the sequence of partial sums $\Sigma a_n c_n$ necessarily converge?

Does it necessarily converge if we let $c_k \in \mathbb{R}$ instead of positive reals?

#Calculus

Easy Math Editor

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Comments

The answer to both questions is no: Let $a_k = (-1)^{k+1},$ and let $c_k = \begin{cases} \frac4{k+1} & k \text{ odd} \\ \frac2{k} & k \text{ even} \end{cases}$ Then $\sum a_n c_n = 2 - 1 + 1 - \frac12 + \frac23 - \frac13 + \frac12 - \frac14 + \cdots$ diverges (every other partial sum is a partial sum of the harmonic series).

You can make the conclusion true if you assume in addition that $c_k$ is monotone, in which case the result is a famous theorem of Dirichlet.

Patrick Corn - 2 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$