How do you prove that the limit $n \to \infty \sum e^{- \frac{n}{10}} cos (\frac{n}{10})$ $n to oo sum e^(-n/10) cos(n/10)$ , for n from 0 to n, is 10.5053, nearly.?

Question

How do you prove that the limit $n \to \infty \sum e^{- \frac{n}{10}} cos (\frac{n}{10})$ $n to oo sum e^(-n/10) cos(n/10)$ , for n from 0 to n, is 10.5053, nearly.?

1 Answer

Impact of this question

1350 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-12-03T19:51:55

See below.

Explanation:

$e^{- \frac{k}{10}} (cos (- \frac{k}{10}) + i sin (- \frac{k}{10})) = e^{- \frac{k}{10}} e^{- i \frac{k}{10}} = e^{- (\frac{i + 1}{10}) k}$ $e^(-k/10)(cos(-k/10)+i sin(-k/10))=e^(-k/10)e^(-ik/10) = e^(-((i+1)/10)k)$

so

$\infty \sum k = 0 e^{- \frac{k}{10}} cos (\frac{k}{10}) = R_{e} (\infty \sum k = 0 e^{- (\frac{i + 1}{10}) k})$ $sum_(k=0)^ooe^(-k/10)cos(k/10) = R_e(sum_(k=0)^oo e^(-((i+1)/10)k))$

but $∣ ∣ ∣ e^{- (\frac{i + 1}{10}) k} ∣ ∣ ∣ = e^{- \frac{k}{10}} < 1$ $abs(e^(-((i+1)/10)k))=e^(-k/10) < 1$ so

$\infty \sum k = 0 e^{- (\frac{i + 1}{10}) k} = \frac{0 - 1}{e^{- \frac{i + 1}{10}} - 1}$ $sum_(k=0)^oo e^(-((i+1)/10)k) = (0-1)/(e^(-(i+1)/10)-1)$ so

$\frac{1}{1 - (e^{- \frac{i + 1}{10}})} = \frac{1}{2 - \frac{2 sinh [\frac{1}{10}]}{e^{\frac{1}{10}} - cos (\frac{1}{10})}} - i ⎛ ⎜ ⎝ \frac{e^{\frac{1}{10}} sin (\frac{1}{10})}{1 + e^{\frac{1}{5}} - 2 e^{\frac{1}{10}} cos (\frac{1}{10})} ⎞ ⎟ ⎠$ $1/(1-(e^(-(i+1)/10)))=1/(2 - (2 sinh[1/10])/(e^(1/10) - Cos(1/10)))-i((e^(1/10) Sin(1/10))/(1 + e^(1/5) - 2 e^(1/10) Cos(1/10)))$

and

$R_{e} ⎛ ⎜ ⎝ \frac{1}{1 - (e^{- \frac{i + 1}{10}})} ⎞ ⎟ ⎠ = \frac{1}{2 - \frac{2 sinh [\frac{1}{10}]}{e^{\frac{1}{10}} - cos (\frac{1}{10})}} = 5.508336109787682$ $R_e(1/(1-(e^(-(i+1)/10))))=1/(2 - (2 sinh[1/10])/(e^(1/10) - Cos(1/10)))=5.508336109787682$

Science

Math

Humanities

... and beyond

How do you prove that the limit $n \to \infty \sum e^{- \frac{n}{10}} cos (\frac{n}{10})$ $n to oo sum e^(-n/10) cos(n/10)$ , for n from 0 to n, is 10.5053, nearly.?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you prove that the limit n→∞∑e−n10cos(n10)n to oo sum e^(-n/10) cos(n/10), for n from 0 to n, is 10.5053, nearly.?

1 Answer

Explanation:

Related questions

Impact of this question

How do you prove that the limit $n \to \infty \sum e^{- \frac{n}{10}} cos (\frac{n}{10})$ $n to oo sum e^(-n/10) cos(n/10)$ , for n from 0 to n, is 10.5053, nearly.?