Match each of the following with the correct statement.

Apr 2010
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1
Match each of the following with the correct statement.
C stands for Convergent, D stands for Divergent.

1.

2.

3.

4.

5.
 

Prove It

MHF Helper
Aug 2008
12,883
4,999
Match each of the following with the correct statement.
C stands for Convergent, D stands for Divergent.

1.

2.

3.

4.

5.
1. \(\displaystyle \frac{8 + 9^n}{9^n} = \frac{8}{9^n} + 1\)

So \(\displaystyle \lim_{n \to \infty}\frac{8 + 9^n}{9^n} = \lim_{n \to \infty}\left(\frac{8}{9^n} + 1\right)\)

\(\displaystyle = 1\)

\(\displaystyle > 0\).

So the series is divergent.
 
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Prove It

MHF Helper
Aug 2008
12,883
4,999
Match each of the following with the correct statement.
C stands for Convergent, D stands for Divergent.

1.

2.

3.

4.

5.

4. \(\displaystyle \sum_{n = 1}^{\infty}n\,e^{-n^2} \leq \int_1^{\infty}x\,e^{-x^2}\,dx\)

\(\displaystyle = -\frac{1}{2}\int_1^{\infty}{-2x\,e^{-x^2}\,dx}\)

\(\displaystyle = \lim_{\varepsilon \to \infty}-\frac{1}{2}\left[e^{-x^2}\right]_1^{\varepsilon}\)

\(\displaystyle =\lim_{\varepsilon \to \infty}-\frac{1}{2}\left[e^{-\varepsilon^2} - e^{-1}\right]\)

\(\displaystyle = -\frac{1}{2}\left[0 - e^{-1}\right]\)

\(\displaystyle = \frac{1}{2}\,e^{-1}\).


By the integral test, the series converges.
 
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