Mathematics: Determine whether the series is convergent or divergent by expressing sn as a telescoping sum (as in example 7). ∞ cos 7 n2 − cos 7 (n + 1)2 n = 1

Determine whether the series is convergent or

Ответ:

nmjohnson2025

13.03.2022

Mathematics

Looks like the series is

$\displaystyle\sum_{n=1}^\infty\cos\frac7{n^2}-\cos\frac7{(n+1)^2}$

Consider the -th partial sum of the series:

$\displaystyle\sum_{n=1}^k\cos\frac7{n^2}-\cos\frac7{(n+1)^2}=$ $\left(\cos7-\cos\dfrac7{2^2}\right)+\left(\cos\dfrac7{2^2}-\cos\dfrac7{3^3}\right)+\cdots+\left(\cos\dfrac7{k^2}-\cos\dfrac7{(k+1)^2}\right)$
We can see some cancellation occurring that leaves us with
$\displaystyle\sum_{n=1}^k\cos\frac7{n^2}-\cos\frac7{(n+1)^2}=\cos7-\cos\frac7{(k+1)^2}$
Then as $k\to\infty$ , we have $\dfrac7{(k+1)^2}\to0$ and $\cos0=1$ , so the series converges to $\cos7-1$ .

0,0(0 оценок)

Ответ:

insaneshootermo

09.05.2022

Mathematics

900 cats

Step-by-step explanation:

let 'c' = # cats

8 = 1080

5 c

cross-multiply:

8c = 5400

c = 900

0,0(0 оценок)

Determine whether the series is convergent or divergent by expressing sn as a telescoping sum (as in example 7). ∞ cos 7 n2 − cos 7 (n + 1)2 n = 1

Ответ:

Ответ:

Ask your question to the AI advisor

More tips

Answers on questions: Mathematics

Ask an AI advisor a question