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?

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- Jan 23rd 2006, 01:39 PMhongster5help 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 PMThePerfectHackerQuote:

Originally Posted by**hongster5**

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 PMhongster5
thx alot. i wasnt thinking of geometric sequences... not that i would have known that equation anyways, lol

- Jan 23rd 2006, 03:52 PMThePerfectHacker
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