Solution Found!
Infinite products An infinite product P = a1 a2a3 …, which
Chapter 12, Problem 76AE(choose chapter or problem)
Infinite products An infinite product \(P=a_{1} a_{2} a_{3} \ldots\), which is denoted \(\prod_{k=1}^{\infty} a_{k}\), is the limit of the sequence of partial products \(\left\{a_1,\ a_1a_2,\ a_1a_2a_3,\ldots\right\}\)
a. Show that the infinite product converges (which means its sequence of partial products converges) provided the series \(\sum_{k=1}^{\infty} \ln a_{k}\) converges.
b. Consider the infinite product
\(P=\prod_{k=2}^{\infty}\left(1-\frac{1}{k^{2}}\right)=\frac{3}{4} \cdot \frac{8}{9} \cdot \frac{15}{16} \cdot \frac{24}{25} \cdots\)
Write out the first few terms of the sequence of partial products,
\(P_{n}=\prod_{k=2}^{n}\left(1-\frac{1}{k^{2}}\right)\)
(for example, \(P_2=\frac{3}{4},\ P_3=\frac{2}{3}\)). Write out enough terms to determine the value of the product, which is \(\lim_{n\rightarrow\infty}\ P_n\).
c. Use the results of parts (a) and (b) to evaluate the series
\(\sum_{k=2}^{\infty} \ln \left(1-\frac{1}{k^{2}}\right)\).
Questions & Answers
QUESTION:
Infinite products An infinite product \(P=a_{1} a_{2} a_{3} \ldots\), which is denoted \(\prod_{k=1}^{\infty} a_{k}\), is the limit of the sequence of partial products \(\left\{a_1,\ a_1a_2,\ a_1a_2a_3,\ldots\right\}\)
a. Show that the infinite product converges (which means its sequence of partial products converges) provided the series \(\sum_{k=1}^{\infty} \ln a_{k}\) converges.
b. Consider the infinite product
\(P=\prod_{k=2}^{\infty}\left(1-\frac{1}{k^{2}}\right)=\frac{3}{4} \cdot \frac{8}{9} \cdot \frac{15}{16} \cdot \frac{24}{25} \cdots\)
Write out the first few terms of the sequence of partial products,
\(P_{n}=\prod_{k=2}^{n}\left(1-\frac{1}{k^{2}}\right)\)
(for example, \(P_2=\frac{3}{4},\ P_3=\frac{2}{3}\)). Write out enough terms to determine the value of the product, which is \(\lim_{n\rightarrow\infty}\ P_n\).
c. Use the results of parts (a) and (b) to evaluate the series
\(\sum_{k=2}^{\infty} \ln \left(1-\frac{1}{k^{2}}\right)\).
ANSWER:
Problem 76 AEInfinite products An infinite product P = a1 a2a3 …, which is denoted , is the limit of the sequence of partial products {a1,.a1 a2,.a1 a2 a3, …}.a. Show that the infinite product converges (which means its sequence of partial products converges)provided the series ln ak converges.b. Consider the infinite product Write out the first few terms of the sequence of partial products. (for example, ). Write out enough terms to determine the value of the product, which is .c. Use the results of parts (a) and (b) to evaluate the series Answer; Step-1; Given, Infinite products ; P = which is denoted , is the limit of the sequence of partial products { a) Now , we have to show that the infinite product converges (which means its sequence of partial products converges) provided the series ln() converges. That is , The product of positive real numbers converges to a nonzero real number if and only if the sum ln() converges . This allows the translation of convergence criteria for infin