Sum of Infinite Series | Calculate the Sum of a Geometric Series

In summary, the first step is to find a Taylor series for F(x). Once you have that, you can use integration to get the solution.
  • #1
Kqwert
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Homework Statement


Find the sum of the series
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Homework Equations

The Attempt at a Solution


Not sure exactly where to start. If I move 3 outside the sum I'm left with 3*sigma(1/n*4^n), which I can rewrite to 3*sigma((1/n)*(1/4)^n), which party looks like a geometric series..Any tips?
 

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  • #2
No, it's not a geometric series. The characteristic of a geometric series is that the ratio between successive terms is a constant. The ratio ##a_{n+1}/a_n## in this case is ##(n 4^n) / [(n+1)4^{n+1}]## or ##n/[4(n+1)]##, definitely not a constant.

That said, I'm not sure off-hand what trick might work here. Here's a PDF that shows some of the tricks for transforming infinite series into something that you can work with.
http://web.math.ucsb.edu/~cmart07/Evaluating Series.pdf

I think the basic hope here would be if you can use the Taylor series approach: Can you transform this into some known Taylor series, evaluated at a particular value? Then it would be that function at that value. Or perhaps the derivative of a known Taylor series? That's where I'd start looking, but I have no specific advice to give you.

Edit: I think I found one. Look at a table of common Taylor series and find one where the n-th term has an n in the denominator. You can make that work for you. That is, the n-th term of the expansion of ##f(x)## can be the same as the n-th term of this series, for a particular x.
 
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  • #3
Kqwert said:

Homework Statement


Find the sum of the series
View attachment 234061

Homework Equations

The Attempt at a Solution


Not sure exactly where to start. If I move 3 outside the sum I'm left with 3*sigma(1/n*4^n), which I can rewrite to 3*sigma((1/n)*(1/4)^n), which party looks like a geometric series..Any tips?

You need to figure out what is
$$F(x) = \sum_{n=1}^\infty \frac{x^n}{n}.$$ This is actually a well-known series that appears in all kinds of calculus textbooks and which you might have see already in your studies. Even if you do not recognize it, there are some standard tricks that people use when facing such series, such as looking at the series for ##dF(x)/dx## or ##\int F(x) \, dx##.
 
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  • #4
When you use integration you can use definite integration (instead of indefinite) and integrate from 0 to t which essentially will make the constant of integration zero.
In the left hand side you ll have

$$\int_0^t \sum (\frac{x^n}{n})'dx=\sum \frac{t^n}{n} -\sum \frac{0^n}{n}$$

and in the right hand side

$$\int_0^t\frac{1}{1-x}dx=-ln(1-t)+ln(1-0)$$
 
  • #5
I use $$log(x)=\sum_{n=1}^{\infty}\frac{1}{n}\left (\frac {x-1}{x}\right )^n$$ for ## x>\frac {1}{2}## and set
##\frac {(x-1)}{x}=\frac {1}{4}## so ##x=\frac {4}{3}##
 

FAQ: Sum of Infinite Series | Calculate the Sum of a Geometric Series

What is the concept of "sum of infinite series"?

The sum of infinite series refers to the sum of an infinite number of terms in a mathematical series, where each term is added to the previous one in a specific pattern.

What is the difference between a finite and infinite series?

A finite series has a limited number of terms, while an infinite series has an infinite number of terms.

How do you determine the convergence or divergence of an infinite series?

The convergence or divergence of an infinite series can be determined by evaluating the limit of the series as the number of terms approaches infinity. If the limit exists and is a finite number, the series is convergent. If the limit does not exist or is infinite, the series is divergent.

What is the formula for finding the sum of an infinite geometric series?

The formula for finding the sum of an infinite geometric series is S = a / (1 - r), where a is the first term and r is the common ratio between terms.

What are some real-world applications of infinite series?

Infinite series are used in various fields of science, such as physics, engineering, and economics, to model and predict real-world phenomena. They are also used in computer science for algorithms and data analysis.

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