Shifted p-series Consider the sequence {Fn}defined by for

Chapter 10, Problem 60AE

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QUESTION:

Shifted p-series Consider the sequence \(\left\{F_{n}\right\}\) defined by

                                \(F_{n}=\sum_{k=1}^{\infty} \frac{1}{k(k+n)}\),

for n = 0, 1, 2, .... When n = 0, the series is a p-series, and we have \(F_{0}=\pi^{2} / 6\) (Exercises 57 and 58).

a. Explain why \(\left\{F_{n}\right\}\) is a decreasing sequence.

b. Plot approximations to \(\left\{F_{n}\right\}\) for n = 1, 2, . . ., 20.

c. Based on your experiments, make a conjecture about \(\lim_{n\rightarrow\infty}\ F_n\).

Questions & Answers

QUESTION:

Shifted p-series Consider the sequence \(\left\{F_{n}\right\}\) defined by

                                \(F_{n}=\sum_{k=1}^{\infty} \frac{1}{k(k+n)}\),

for n = 0, 1, 2, .... When n = 0, the series is a p-series, and we have \(F_{0}=\pi^{2} / 6\) (Exercises 57 and 58).

a. Explain why \(\left\{F_{n}\right\}\) is a decreasing sequence.

b. Plot approximations to \(\left\{F_{n}\right\}\) for n = 1, 2, . . ., 20.

c. Based on your experiments, make a conjecture about \(\lim_{n\rightarrow\infty}\ F_n\).

ANSWER:

Solution:-Step1Given that for n = 0, 1, 2… When n = 0, the series is a p-series, and we have F0 = 2/6 Step2To finda. Explain why {Fn}is a decreasing sequence.__

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