6.2 Properties of power series  (Page 7/10)

$\frac{30}{\left(x^2+1\right)\left(x^2-9\right)}$

${\displaystyle \sum _{n=1}^{\infty }\frac{1}{x^n}}$

${\displaystyle \sum _{n=1}^{\infty }\frac{1}{x^{2n}}}$

${\displaystyle \sum _{n=1}^{\infty }\frac{1}{{\left(x-3\right)}^{2n-1}}}$

${\displaystyle \sum _{n=1}^{\infty }\left(\frac{1}{{\left(x-3\right)}^{2n-1}}-\frac{1}{{\left(x-2\right)}^{2n-1}}\right)}$

Calculate the present values P of an annuity in which $10,000 is to be paid out annually for a period of 20 years, assuming interest rates of $r=0.03,r=0.05,$ and $r=0.07.$

Calculate the present values P of annuities in which $9,000 is to be paid out annually perpetually, assuming interest rates of $r=0.03,r=0.05$ and $r=0.07.$

Calculate the annual payouts C to be given for 20 years on annuities having present value $100,000 assuming respective interest rates of $r=0.03,r=0.05,$ and $r=0.07.$

Calculate the annual payouts C to be given perpetually on annuities having present value $100,000 assuming respective interest rates of $r=0.03,r=0.05,$ and $r=0.07.$

Suppose that an annuity has a present value $P=1\text{million dollars}.$ What interest rate r would allow for perpetual annual payouts of $50,000?

Suppose that an annuity has a present value $P=10\text{million dollars}\text{.}$ What interest rate r would allow for perpetual annual payouts of $100,000?

$x+x^2-x^3+x^4+x^5-x^6+\text{⋯}$ ( Hint: Group powers x ^{3 k} , $x^{3k-1},$ and $x^{3k-2}.)$

$x+x^2-x^3-x^4+x^5+x^6-x^7-x^8+\text{⋯}$ ( Hint: Group powers x ^{4 k} , $x^{4k-1},$ etc.)

$x-x^2-x^3+x^4-x^5-x^6+x^7-\text{⋯}$ ( Hint: Group powers x ^{3 k} , $x^{3k-1},$ and $x^{3k-2}.)$

$\frac{x}{2}+\frac{x^2}{4}-\frac{x^3}{8}+\frac{x^4}{16}+\frac{x^5}{32}-\frac{x^6}{64}+\text{⋯}$ ( Hint: Group powers ${\left(\frac{x}{2}\right)}^{3k},{\left(\frac{x}{2}\right)}^{3k-1},$ and ${\left(\frac{x}{2}\right)}^{3k-2}.)$

$f\left(x\right)=2{\displaystyle \sum _{n=0}^{\infty }{x}^n},g\left(x\right)={\displaystyle \sum _{n=0}^{\infty }n{x}^n}$

$f\left(x\right)={\displaystyle \sum _{n=1}^{\infty }{x}^n},g\left(x\right)={\displaystyle \sum _{n=1}^{\infty }\frac{1}{n}{x}^n}.$ Express the coefficients of $f\left(x\right)g\left(x\right)$ in terms of $H_n={\displaystyle \sum _{k=1}^n\frac{1}{k}}.$

$f\left(x\right)=g\left(x\right)={\displaystyle \sum _{n=1}^{\infty }{\left(\frac{x}{2}\right)}^n}$

$f\left(x\right)=g\left(x\right)={\displaystyle \sum _{n=1}^{\infty }n{x}^n}$

$f\left(x\right)=\frac{1}{1+x}={\displaystyle \sum _{n=0}^{\infty }{\left(\mathrm{-1}\right)}^n{x}^n}$

$f\left(x\right)=\frac{1}{1-x^2}={\displaystyle \sum _{n=0}^{\infty }{x}^{2n}}$

$f\left(x\right)=\frac{2x}{{\left(1+x^2\right)}^2}={\displaystyle \sum _{n=1}^{\infty }{\left(\mathrm{-1}\right)}^n\left(2n\right)x^{2n-1}}$

$f\left(x\right)=\frac{2x}{1+x^2}=2{\displaystyle \sum _{n=0}^{\infty }{\left(\mathrm{-1}\right)}^n{x}^{2n+1}}$

Evaluate ${\displaystyle \sum _{n=1}^{\infty }\frac{n}{2^n}}$ as $f^{\prime }\left(\frac{1}{2}\right)$ where $f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }{x}^n}.$