Finding sum of infinite geometric series

In summary, the conversation discussed finding the sum of an infinite geometric series with terms 1 - √2 + 2 - 2√2 + ... The expert calculated the common difference and determined that the answer is A. However, it was also mentioned that for a geometric series to converge, the absolute value of the common ratio must be less than 1. Since the common ratio in this series is 2, the series diverges.
  • #1
fluffertoes
16
0
find the sum of this infinite geometric series:
1 - √2 + 2 - 2√2 + ...

a.) .414
b.) -2.414
c.) series diverges
d.) 2

I found that the common difference is 2, so I calculated this:

S= -.414/-1
s= .414

So i got that the answer is A, but will you check this?
 
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  • #2
In order for this to be a geometric series, we must have:

\(\displaystyle S=(1-\sqrt{2})\sum_{k=0}^{\infty}\left(a^k\right)\)

Going by the terms given, it appears that $a=2$, and so what does this tell us about the convergence?
 
  • #3
MarkFL said:
In order for this to be a geometric series, we must have:

\(\displaystyle S=(1-\sqrt{2})\sum_{k=0}^{\infty}\left(a^k\right)\)

Going by the terms given, it appears that $a=2$, and so what does this tell us about the convergence?

oh gosh i don't know
 
  • #4
fluffertoes said:
oh gosh i don't know

Recall that:

\(\displaystyle \sum_{k=0}^{\infty}r^k=\frac{1}{1-r}\) but only if $|r|<1$

Otherwise, the sum diverges. :D
 
  • #5
MarkFL said:
Recall that:

\(\displaystyle \sum_{k=0}^{\infty}r^k=\frac{1}{1-r}\) but only if $|r|<1$

Otherwise, the sum diverges. :D

But r stood to equal 2, so the answer should be c?
 
  • #6
fluffertoes said:
but r stood to equal 2, so the answer should be c?

hello??
 
  • #7
fluffertoes said:
But r stood to equal 2, so the answer should be c?

Yes.

fluffertoes said:
hello??

Sorry, but I was working on a coding request at vBorg. (Sweating)
 

Related to Finding sum of infinite geometric series

1. What is an infinite geometric series?

An infinite geometric series is a series of numbers that follows a specific pattern, where each term is multiplied by a fixed number called the common ratio. The series continues infinitely, with each term becoming smaller and smaller.

2. How do you find the sum of an infinite geometric series?

To find the sum of an infinite geometric series, you can use the formula S = a / (1 - r), where "a" is the first term of the series and "r" is the common ratio. This formula only works when the absolute value of the common ratio, |r|, is less than 1. If |r| is greater than or equal to 1, the series does not have a sum.

3. What is the significance of the common ratio in an infinite geometric series?

The common ratio determines the behavior of the series. If the absolute value of the common ratio is less than 1, the series will approach a finite sum. If the common ratio is equal to 1, the series will have a sum of infinity. And if the absolute value of the common ratio is greater than 1, the series will diverge and not have a sum.

4. Can an infinite geometric series have a negative sum?

Yes, an infinite geometric series can have a negative sum if the absolute value of the common ratio is between 0 and 1. In this case, the series will approach a negative value as the number of terms increases.

5. How is the sum of an infinite geometric series used in real-world applications?

The concept of an infinite geometric series is used in various fields such as mathematics, physics, and finance. For example, it can be used to calculate the total distance traveled by an object that is moving at a decreasing speed, or to determine the total interest earned on a loan with a fixed interest rate. It is also used in computer science and engineering for calculations involving infinite loops.

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