help with infinite sum

**hongster5** · January 23rd 2006, 01:39 PM

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?

**ThePerfectHacker** · January 23rd 2006, 01:58 PM

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,
$1+t+t^2+t^3+...=5$

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

Something interesting to note,
the eqaution
$\frac{1}{1-t}=r$
Always has a unique solution,
namely,
$\frac{r-1}{r}$
and that,
$|\frac{r-1}{r}|<1$
Thus, any infinite geometric sum can be made to converge to any real number.

**hongster5** · January 23rd 2006, 03:35 PM

thx alot. i wasnt thinking of geometric sequences... not that i would have known that equation anyways, lol

**ThePerfectHacker** · January 23rd 2006, 03:52 PM

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