1. ## power series

Use series you know to show that:

1 - 1/3 + 1/5 - 1/7 + ... = pi/4

I get that arctanx = x - 1/3x^3 + 1/5x^5 - 1/7x^7
and arctan 1 = pi/4

but I'm not sure how I'm supposed to show this; what is the connection? Is this the connection I am looking for?

2. Originally Posted by sprinks
Use series you know to show that:

1 - 1/3 + 1/5 - 1/7 + ... = pi/4

I get that arctanx = x - 1/3x^3 + 1/5x^5 - 1/7x^7
and arctan 1 = pi/4

but I'm not sure how I'm supposed to show this; what is the connection? Is this the connection I am looking for?
Originally Posted by sprinks
Use series you know to show that:

1 - 1/3 + 1/5 - 1/7 + ... = pi/4

I get that arctanx = x - 1/3x^3 + 1/5x^5 - 1/7x^7
and arctan 1 = pi/4

but I'm not sure how I'm supposed to show this; what is the connection? Is this the connection I am looking for?
$\displaystyle \frac{1}{1+x^2} = 1 - x^2 + x^4 - x^6 + ...$ for $\displaystyle |x|<1$.

Integrate from $\displaystyle 0$ to $\displaystyle t$ where $\displaystyle |t|<1$ we get:
$\displaystyle \int_0^t \frac{dx}{1+x^2} = \int_0^t 1 - x^2 + x^4 - ... dx$
Because of uniform convergence we can integrate term-by-term.
Thus,
$\displaystyle \tan^{-1} t = t - \frac{t^3}{3} + \frac{t^5}{5} - ...$ for $\displaystyle |t|<1$.
Now this series converges for $\displaystyle t=1$ so by applying "Abel's theorem" we can evalute this even at $\displaystyle t=1$ so we get:
$\displaystyle \frac{\pi}{4} = 1 - \frac{1}{3}+ ...$