Question

(Note: For Problems 6 and 7, you will need to prove the sequence of the partial sums {sm} is convergent or divergent. Do not just quote any tests for infinite series.) 6. Prove that the infinite series $\sum_{n=1}^{\infty} \frac{1}{n(n+2)}$ is convergent, and then find the value of it.

Name: note for problems 6 and 7 you will need to prove the sequence of the partial sums sm is convergent or divergent do not just quote any tests for infinite series 6 prove that the infinite seri 94281
Uploaded: 2023-04-05T11:37:43-08:00
Duration: 4 min 41 s
Channel: Audrey Fong
Description: note for problems 6 and 7 you will need to prove the sequence of the partial sums sm is convergent or divergent do not just quote any tests for infinite series 6 prove that the infinite seri 94281

          (Note: For Problems 6 and 7, you will need to prove the sequence of the partial sums {sm} is
convergent or divergent. Do not just quote any tests for infinite series.)
6. Prove that the infinite series $\sum_{n=1}^{\infty} \frac{1}{n(n+2)}$ is convergent, and then find the value of it.

note for problems 6 and 7 you will need to prove the sequence of the partial sums sm is convergent or divergent do not just quote any tests for infinite series 6 prove that the infinite seri 94281

Added by Jose Angel S.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Audrey Fong

04/05/2023

Step 1

$$\frac{1}{n(n+2)} = \frac{A}{n} + \frac{B}{n+2}$$ Multiplying both sides by $n(n+2)$, we get $$1 = A(n+2) + Bn$$ If $n=0$, then $1 = 2A$, so $A = \frac{1}{2}$. If $n=-2$, then $1 = -2B$, so $B = -\frac{1}{2}$. Thus, $$\frac{1}{n(n+2)} = \frac{1}{2} \left( Show more…

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(Note: For Problems 6 and 7, you will need to prove the sequence of the partial sums {sm} is convergent or divergent. Do not just quote any tests for infinite series.) 6. Prove that the infinite series $\sum_{n=1}^{\infty} \frac{1}{n(n+2)}$ is convergent, and then find the value of it.