What is the sum of the infinite series: log(1-1/(n+1)^2)?

In summary, the conversation discusses a series and how to determine its sum. It mentions using techniques such as checking for a geometric or telescoping series, and eventually the poster realizes it can be simplified into a single ratio. The conversation ends with them thanking each other for their help.
  • #1
kidmode01
53
0

Homework Statement


Determine the sum of the following series:

[tex]\sum_{n=1}^{inf} log(1-1/(n+1)^2)[/tex]

Sorry for poor latex, that is supposed to say infinity.

Homework Equations


How might we turn this into an easier function to deal with?


The Attempt at a Solution



So far I've only proved convergence of the series. I'm not really sure where to begin. Any help is appreciated. I've ran it through maple and come out with the sum equaling -ln(2).

I thought of trying to work backwards to see if this series was some sort of taylor expansion but I failed at that. I just don't see any elementary techniques. Obviously maple spotted something I haven't lol.
 
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  • #2
Maybe it is a geometric series. If so, why not figure out the 1st and 2nd term and the find the r. You are given the first term then use the formula a/ (1-r) where a is the first term and r is the rate.
 
  • #3
Okay,

So I worked out the first two terms:

A1 = log(1-1/4) = log(3/4)

A2 = log(1-1/9) = log(8/9)

then r = log(8/9) / log(3/4)

But then checking A3 = log(1-1/16),

A3 does not equal r*A2.

So this is not a geometric series?
 
  • #4
It's not geometric. Your last hope is that it is a telescoping series. Is it? Write 1-1/(n+1)^2 as a single ratio and factor it up cleverly.
 
  • #5
kidmode01 said:
So this is not a geometric series?
Defenitely not.

Try this instead:


[tex]\sum ln\left(1-\frac{1}{(n+1)^2}\right)=\sumln\left(\frac{(n+1)^2-1}{(n+1)^2}\right)=...=[/tex]

[tex]=\sum ln\keft(\frac{n(n+2)}{(n+1)^2}\left)=...=\sum[ln(n)+ln(n+2)-2ln(n+1)]=[/tex]

[tex] =\sum[ln(n+2)-ln(n+1)]+\sum [ln(n)-ln(n+1)][/tex]

Now stuff will cancel out, i already did more than i was supposed to.

Edit: Sorry Dick, i didn't know you were already on it!
 
  • #6
sutupidmath said:
Now stuff will cancel out, i already did more than i was supposed to.

Edit: Sorry Dick, i didn't know you were already on it!

S'ok. How could you, I hadn't posted yet. But you should start with a hint. You shouldn't steal all the fun from what seems to be a pretty clever poster. I consider 'pretty clever' to be realizing and proving that a series is not geometric.
 
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  • #7
Haha, well thanks a lot guys :) I had thought of building it into a single ratio but I thought I was only further complicating it. Thanks again.
 

What is a sum of an infinite series?

The sum of an infinite series is the total value obtained by adding an infinite number of terms together. This is a concept often used in mathematics and physics to represent the total value of a sequence that continues infinitely.

How is the sum of an infinite series calculated?

The sum of an infinite series can be calculated using various methods, such as the geometric series formula, telescoping series, or the ratio test. It is important to note that not all infinite series have a finite sum, and some may diverge to infinity.

What is the significance of the sum of an infinite series?

The sum of an infinite series is often used to represent real-life phenomena, such as the growth of a population or the value of an investment over time. It also plays a crucial role in many mathematical and scientific calculations and theories.

What is the difference between a convergent and divergent infinite series?

A convergent infinite series is one in which the sum of the terms approaches a finite value as the number of terms increases. In contrast, a divergent infinite series is one in which the sum of the terms either approaches infinity or does not approach any value at all.

Are there any real-world applications of the sum of an infinite series?

Yes, the concept of the sum of an infinite series is widely used in various fields, such as physics, economics, and engineering. It is often used to model natural phenomena, make predictions, and solve problems involving continuous growth or decay.

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