Solution Found!
Shifted p-series Consider the sequence {Fn}defined by for
Chapter 10, Problem 60AE(choose chapter or problem)
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.__