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Starting with the geometric series $\sum _{n=0}^{\mathrm{\infty }}{x}^n$, find the sum of the series
$\sum _{n=1}^{\mathrm{\infty }}n{x}^{n-1},|x|<1$

Determine if a geometric series converges or diverges
$10-4+1.6-0.64+...$ 
If it convergent, find the sum.

Missing number in the series $9,?,6561,43046721$ is: $8125623118$

Find the sim of each of the following series.
1) $\sum _{n=1}^{\mathrm{\infty }}n{x}^n,|x|<1$
2) $\sum _{n=1}^{\mathrm{\infty }}\frac{n}{8^n}$

Let P(k) be a statement that $\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+...+\frac{1}{k\cdot (k+1)}=$
for: The basis step to prove $P(k)$ is that at $k=1,?$ is true.
for:Show that $P(1)$ is true by completing the basis step proof. Left side of $P(k)$ and Right side of $P(k)$
for: Identify the inductive hypothesis used to prove $P(k)$.
for: Identify the inductive step used to prove $P(k+1).$

Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series.
${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{1}{n+3}}$

Find the power series representation for g centered at 0 by differentiating or integrating the power series for f(perhaps more than once). Give the interval of convergence for the resulting series.
$g(x)=\ln (1-2x)$ using $f(x)=\frac{1}{1}-2x$

Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series
$(1+x{)}^{-2}=1-2x+3x^2-4x^3+\cdots for-1$
$(1+4x{)}^{-2}$

Q1. Does a series ${\displaystyle \sum _{\mathrm{\infty }}^{n=1}bn}$ converge if $bn\to 0$? Justify your answer by at least two examples?

Write the following arithmetic series in summation notation.
$8+10+12+\cdots +34$