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Magazine Problem 87
State whether the given series converges and explain why. \(\sum_{n=1}^{\infty} \frac{1}{n+1000}\)
Problem 88
State whether the given series converges and explain why. \(\sum_{n=1}^{\infty} \frac{1}{n+10^{80}}\)
Problem 89
State whether the given series converges and explain why. \(1+\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\cdots\)
Problem 90
State whether the given series converges and explain why. \(1+\frac{e}{\pi}+\frac{e^{2}}{\pi^{2}}+\frac{e^{3}}{\pi^{3}}+\cdots\)
Problem 91
State whether the given series converges and explain why. \(1+\frac{\pi}{e}+\frac{\pi^{2}}{e^{4}}+\frac{\pi^{3}}{e^{6}}+\frac{\pi^{4}}{e^{8}}+\cdots\)
Problem 94
For \(a_{n}\) as follows, write the sum as a geometric series of the form \(\sum_{n=1}^{\infty} a r^{n} .\) State whether the series converges and if it does, find the value of \(\sum a_{n}\). $$ a_{1}=2 \text { and } a_{n} / a_{n+1}=1 / 2 \text { for } n \geq 1 $$
Problem 95
For \(a_{n}\) as follows, write the sum as a geometric series of the form \(\sum_{n=1}^{\infty} a r^{n} .\) State whether the series converges and if it does, find the value of \(\sum a_{n}\). $$ a_{1}=10 \text { and } a_{n} / a_{n+1}=10 \text { for } n \geq 1 $$
Problem 97
Use the identity \(\frac{1}{1-y}=\sum_{n=0}^{\infty} y^{n}\) to express the function as a geometric series in the indicated term. $$ \frac{x}{1+x} \text { in } x $$
Problem 98
Use the identity \(\frac{1}{1-y}=\sum_{n=0}^{\infty} y^{n}\) to express the function as a geometric series in the indicated term. \(\frac{\sqrt{x}}{1-x^{3 / 2}}\) in \(\sqrt{x}\)
Problem 99
Use the identity \(\frac{1}{1-y}=\sum_{n=0}^{\infty} y^{n}\) to express the function as a geometric series in the indicated term. \(\frac{1}{1+\sin ^{2} x}\) in \(\sin x\)
