91Ó°ÊÓ

Two complex numbers \(a+i b\) and \(c+i d\) are equal if and only if \(a=c\) and \(b=d .\) Use this fact to evaluate $$\int e^{a x} \cos b x d x \quad \text { and } \int e^{a x} \sin b x d x$$ from $$\int e^{(a+i b) x} d x=\frac{a-i b}{a^{2}+b^{2}} e^{(a+i b) x}+C$$ where \(C=C_{1}+i C_{2}\) is a complex constant of integration.

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$a_{n}=\frac{(10 / 11)^{n}}{(9 / 10)^{n}+(11 / 12)^{n}}$$

Find the values of \(x\) for which the given geometric series converges. Also, find the sum of the series (as a function of \(x\) ) for those values of \(x .\) $$\sum_{n=0}^{\infty}(-1)^{n} x^{-2 n}$$

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$a_{n}=\tanh n$$

Find the values of \(x\) for which the given geometric series converges. Also, find the sum of the series (as a function of \(x\) ) for those values of \(x .\) $$\sum_{n=0}^{\infty}(-1)^{n}(x+1)^{n}$$

Find the values of \(x\) for which the given geometric series converges. Also, find the sum of the series (as a function of \(x\) ) for those values of \(x .\) $$\sum_{n=0}^{\infty}\left(-\frac{1}{2}\right)^{n}(x-3)^{n}$$

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$a_{n}=\sinh (\ln n)$$

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$a_{n}=\frac{n^{2}}{2 n-1} \sin \frac{1}{n}$$

Find the values of \(x\) for which the given geometric series converges. Also, find the sum of the series (as a function of \(x\) ) for those values of \(x .\) $$\sum_{n=0}^{\infty} \sin ^{n} x$$

Find the values of \(x\) for which the given geometric series converges. Also, find the sum of the series (as a function of \(x\) ) for those values of \(x .\) \( \)\sum_{n=0}^{\infty}(\ln x)^{n}$$