can someone walk me through this, step by step please

Question

anonymous · Answer

$$\sum_{n=0}^{\infty} \frac{ 1 }{ (\sqrt11)^n }$$

anonymous · Answer

Find the sum of the series, if it converges

phi · Answer

do you know how to find the sum of a geometric series ? https://en.wikipedia.org/wiki/Geometric_series#Formula

anonymous · Answer

i would really appreciate the help. I am not good with this question at all. and the square root has me thrown off

anonymous · Answer

I know about $$\frac{ a }{ 1-r }$$

anonymous · Answer

and how to test for convergence with limit

anonymous · Answer

and how it is divergent it r is less than -1 or more than 1

phi · Answer

in this case, r= 1/sqrt(11) = 0.30511...

phi · Answer

a is 1

anonymous · Answer

$$\frac{ 1 }{ 1-\frac{ 1 }{ \sqrt11 } }$$ ?

phi · Answer

yes.  we can play with that to make it look nicer, but that is the answer.

anonymous · Answer

do i multiply sqrt 11 to top and bottom so I can subtract the two fractions?

anonymous · Answer

$$\frac{ 1 }{ \frac{ 11 }{  11} -\frac{ \sqrt 11 }{ 11 }}$$

phi · Answer

yes, and that is
$$ \frac{1}{\frac{11- \sqrt{11}}{11}} = \frac{11}{11- \sqrt{11}}$$

phi · Answer

we could also get to
$$ \frac{11+\sqrt{11}}{10} $$

anonymous · Answer

this is where i get really thrown off... how do i simplify this now?

anonymous · Answer

oh.... the ten... how did you do that?

phi · Answer

$$ \frac{11}{11- \sqrt{11}} \cdot \frac{11+\sqrt{11}}{11+\sqrt{11}}$$

anonymous · Answer

this would then be $$\frac{ 1 }{ 10 }(11+\sqrt 11)$$

phi · Answer

but I would not bother too much.  They all mean the same value, about 1.43166...

anonymous · Answer

hmmm. I thank you kindly sir

phi · Answer

so to answer the question, you know it converges because | r | < 1
and you can use the formula you posted up above to find the sum

anonymous · Answer

the problem is, my professor does not want decimals, he wants the fraction form :(

anonymous · Answer

yes?

anonymous · Answer

i meant yes, no question

phi · Answer

people generally would use the last form then... no square roots in the denominator (a hold-over from the days before calculators when dividing by square roots was too hard)

anonymous · Answer

lol, they are still too hard for me (obviously) :) thanks a bunch