# help with infinite sum

• Jan 23rd 2006, 01:39 PM
hongster5
help with infinite sum
i'm trying to do some calc hw, but i ran into this equation

Σ t^k = 5
k=0

is there some kind of rule which changes this equation so i can solve for k?
• Jan 23rd 2006, 01:58 PM
ThePerfectHacker
Quote:

Originally Posted by hongster5
i'm trying to do some calc hw, but i ran into this equation

Σ t^k = 5
k=0

is there some kind of rule which changes this equation so i can solve for k?

Solve for k?!? That makes no sense you probably mean for t

This is,
$\displaystyle 1+t+t^2+t^3+...=5$

Assuming that this infinite series converges then it is a geometric series for $\displaystyle |t|<1$ and such as its sum is 5. But its sum is given by $\displaystyle \frac{1}{1-t}$.
Thus,
$\displaystyle \frac{1}{1-t}=5$
thus, $\displaystyle t=\frac{4}{5}$
Q.E.D.

Something interesting to note,
the eqaution
$\displaystyle \frac{1}{1-t}=r$
Always has a unique solution,
namely,
$\displaystyle \frac{r-1}{r}$
and that,
$\displaystyle |\frac{r-1}{r}|<1$
Thus, any infinite geometric sum can be made to converge to any real number.
• Jan 23rd 2006, 03:35 PM
hongster5
thx alot. i wasnt thinking of geometric sequences... not that i would have known that equation anyways, lol
• Jan 23rd 2006, 03:52 PM
ThePerfectHacker
Welcome