$$\sum_{K=1}^{\infty} 4(\frac{ 1 }{ 2 })^{k-1}$$

Question

anonymous · Answer

Evaluate the following

jhannybean · Answer

This is a geometric series, right?

anonymous · Answer

Take 4 outside.
It becomes an infinite GP with common ratio 1/2

zzr0ck3r · Answer

what is the common ratio?

zzr0ck3r · Answer

what is the first term?, what do we do when we know these two things?

anonymous · Answer

First term is 1

anonymous · Answer

wait i thought first term was 4?

jhannybean · Answer

yes, it is.

anonymous · Answer

Common Ratio 1/2... Calculate sum
And multiply the sum by 4 ! 4 is isolated.

luigi0210 · Answer

4(1/2)^0
4(1)
=4

zzr0ck3r · Answer

factor out the 4

jhannybean · Answer

$$\large \sum_{n=1}^{\infty}a_{1}r^{n-1}$$ so a1 =4.

anonymous · Answer

answer would be 8 !! (Final Answer)

anonymous · Answer

The answer is supposed to be 8 but i dont quite get how you get taht

anonymous · Answer

Factor out 4.. and forget it
Calculate Sum 1 + 1/2 + 1/4 + 1/8 ...... upto infinity
This sum is 2..
Multiply by 4 and you get 8

jhannybean · Answer

wait...by "evaluating" are you looking to see if it's convergent/divergent, and if it's convergent,finding the sum?

zzr0ck3r · Answer

factor out the 4, then treat it as (1/2)^n and you have first term is 1
then sum is 1/(1-1/2) = 2 then multiply by 4 and get 8

anonymous · Answer

It's obviously convergent..

anonymous · Answer

pretty sure its convergent

zzr0ck3r · Answer

it is, because (1/2) <1

anonymous · Answer

so i use the sum of geometric formula then i multiply by 4?

zzr0ck3r · Answer

yes

anonymous · Answer

YEs ! @Yellowpanda

anonymous · Answer

okay thank you!

anonymous · Answer

welcome :)

zzr0ck3r · Answer

treat it like $$\sum{}{}(\frac{1}{2})^{n}$$

zzr0ck3r · Answer

then multiply by 4 because of ab+ac+ad....=a(b+c+d....

jhannybean · Answer

Ahh, i see.

a1=4, r = 1/2

$$\large S_{n}= \frac{a}{1-r} = \frac{4}{1-(1/2)} = 8$$

jhannybean · Answer

And this only applies because the series is convergent....

zzr0ck3r · Answer

yes, and note, you could have just factored the 4 out, treated the first term as 1, then multiply the 4 back when you are done. That is what we were saying.

zzr0ck3r · Answer

some people get confused on why the ratio is (1/2) when there is a 4 next to it, so I factor it out to show it does not matter.

jhannybean · Answer

Ohh I see, I see.