Question

lim [ 7/10 + 29/100 + .. + (2^n + 5^n) / 10^n ] . x->infinity

Answer 1

\[\frac{2^n+5^n}{10^n}=\frac{2^n}{10^n}+\frac{5^n}{10^n}=\frac{2^n}{2^n\cdot5^n}+\frac{5^n}{2^n\cdot5^n}=\frac{1}{5^n}+\frac{1}{2^n}\]try using this to get an answer.

Answer 2

Ummm.. i\m not quite sure on how to do it .

Answer 3

it comes down to knowing how to manipulate summations. you want:\[\sum_{n=1}^\infty \frac{2^n+5^n}{10^n}\] I just showed this is the same as:\[\sum_{n=1}^\infty \left(\frac{1}{5^n}+\frac{1}{2^n}\right)\]Now we can break the summation over addition, so we get:\[\sum_{n=1}^\infty \frac{1}{5^n}+\sum_{n=1}^\infty \frac{1}{2^n}\]These two are applications of the infinite geometric series formula.

Answer 4

I actually haven\t studied summation , or that E thing yet . Are there any other ways to do it?

Answer 5

Not really =/ do you know what a geometric series is? if I were to write:\[\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+\left(\frac{1}{2}\right)^n+\cdots\] could you tell me what that infinite sum is?

Answer 6

S= a1/q-1 .

Answer 7

Because if you dont know those, there is no way you could get the answer to this question unfortunately, no matter how i word the solution. =/

Answer 8

ah, ok! thats good :)

Answer 9

:)

Answer 10

So you know that if i have an infinite geometric series, the sum is the first term divided by 1- q, where q is whats called the "common ratio"

In the example i gave with 1/2's, the sum would be:\[\frac{\frac{1}{2}}{1-\frac{1}{2}}=\frac{\frac{1}{2}}{\frac{1}{2}}=1\]because a1= 1/2, and q = 1/2. So now lets look at your problem.

Answer 11

Because I showed that:\[\frac{2^n+5^n}{10^n}=\frac{1}{5^n}+\frac{1}{2^n}\], we can rewrite your series:\[\frac{7}{10}+\frac{29}{100}+\cdots+\frac{2^n+5^n}{10^n}+\cdots\]
as:
\[(\frac{1}{5}+\frac{1}{2})+(\frac{1}{25}+\frac{1}{4})+\cdots+(\frac{1}{5^n}+\frac{1}{2^n})+\cdots\]rearranging the terms we obtain:\[(\frac{1}{5}+\frac{1}{25}+\cdots+\left(\frac{1}{5}\right)^n+\cdots)+(\frac{1}{2}+\frac{1}{4}+\cdots+\left(\frac{1}{2}\right)^n+\cdots)\] The left term is an infinite series with a1= 1/5 and q = 1/5. the second term is another infinite series with a1=1/2 and q = 12 (which we already showed had a sum of 1.) that should take care of your problem.

Answer 12

And the answer is 1.25?

Answer 13

5/4 = 1.25