91Ó°ÊÓ

For Exercises \(19-24,\) consider a geometric sequence with first term \(b\) and ratio \(r\) of consecutive terms. (a) Write the sequence using the three-dot notation, giving the first four terms. (b) Give the \(100^{\text {th }}\) term of the sequence. $$ b=4, r=-5 $$

Express $$ 5.1372647264 \ldots $$ as a fraction; here the digits 7264 repeat forever.

Evaluate the geometric series. $$ \frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\cdots+\frac{1}{4^{50}} $$

For Exercises \(19-24,\) consider a geometric sequence with first term \(b\) and ratio \(r\) of consecutive terms. (a) Write the sequence using the three-dot notation, giving the first four terms. (b) Give the \(100^{\text {th }}\) term of the sequence. $$ b=2, r=\frac{1}{3} $$

Evaluate \(\lim _{n \rightarrow \infty} n \sin \frac{1}{n}\)

For Exercises \(19-24,\) consider a geometric sequence with first term \(b\) and ratio \(r\) of consecutive terms. (a) Write the sequence using the three-dot notation, giving the first four terms. (b) Give the \(100^{\text {th }}\) term of the sequence. $$ b=5, r=\frac{2}{3} $$

Evaluate \(\lim _{n \rightarrow \infty} n \tan \frac{1}{n}\).

Evaluate the geometric series. $$ \frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\cdots+\frac{1}{3^{33}} $$

Evaluate \(\lim _{n \rightarrow \infty} n \ln \left(1+\frac{2}{n}\right)\).

Evaluate the geometric series. $$ 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots+\frac{1}{2^{80}}-\frac{1}{2^{81}} $$