Summation Problem 6

Algebra Level 3

Find the value of ∑ x = 1 ∞ 1 e x \displaystyle\sum_{x=1}^{\infty}\frac1{e^x} up to 2 decimal places.


The answer is 0.58.

Shashank Rustagi by Shashank Rustagi , 24, India

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3 solutions

∑ x = 1 ∞ e − x = ∑ x = 1 ∞ 1 e x = ∑ x = 1 ∞ 1 e × 1 e x − 1 = 1 e × ∑ x = 1 ∞ 1 e x − 1 = 1 e × 1 1 − 1 e = 1 e − 1 ≈ 0.58 \sum_{x=1}^∞e^{-x}=\sum_{x=1}^∞\frac{1}{e^x}=\sum_{x=1}^∞\frac{1}{e}×\frac{1}{e^{x-1}}=\frac{1}{e}×\sum_{x=1}^∞\frac{1}{e}^{x-1}=\frac{1}{e}×\frac{1}{1-\frac{1}{e}}=\frac{1}{e-1}≈\boxed{0.58}

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Atul Shivam
Oct 28, 2015

Simply apply g.p to get sum for infinite terms we have s = 1 e 1 − 1 e s=\frac{\frac{1}{e}}{1-\frac{1}{e}} which is equal to 0.58197671 \boxed{0.58197671}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

How to apply gp?

Debmeet Banerjee - 5 years, 7 months ago
Shashank Rustagi
Jun 17, 2015

Apply GP and for infinite sum we get 0.58

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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