Find the Summation Notation and the Radius of Convergence

In summary: Also, I'm not familiar with the concept of "radius of convergence".In summary, the conversation discusses finding the summation notation and radius of convergence of a given series. The equation for the summation is derived by analyzing the pattern of alternating values in the given sequence. It is also determined that the series does not have a radius of convergence as it is not a power series.
  • #1
Mastur
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Homework Statement


Find the Summation Notation and Radius of Convergence of this series.

5, x, 10, x, ...

The Attempt at a Solution


I don't know how did they come up with that equation.. But the summation seems right.. Can anyone tell me how did they arrive with that equation? I've tried using the common ratio but I don't know what to do after having the ratio..

And for the radius of convergence, its L = lim (An+1)/(An) as n=> infinity..

Using the summation notation, I ended up infinityinfinity over infinityinfinity. I thought its equal to 1.. ~_~

Sorry, seems like the latex codes now isn't the same as the latex codes before. I don't know how to use the new latex codes..
 

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  • #2
If n is odd, [itex](-1)^n= -1[/itex] and if n is even, [itex](-1)^n= 1[/itex]. That means that if n is odd [itex](1+ (-1)^n)/2= 0[/itex] and if n is even, [itex](1+ (-1)^n)/2= 1[/itex]. Of course, if whether n is even or odd, n+ 1 is the opposite.

So for any odd n, [itex]5^{(1+(-1)^n)/2}= 5^0= 1[/itex] and [itex]x^{(1+ (-1)^{n+1})/2}= x^1= x[/itex] while for n even, [itex]5^{(1+(-1)^n)/2}= 5^1= 5[/itex] and [itex]x^{(1+ (-1)^{n+1})/2}= x^0= 1[/itex]. That is, [itex]5^{(1+(-1)^n)/2}x^{(1+(-1)^n)/2}[/itex] just alternates 5 and x as your sequence does.

As for finding the "radius of convergence", it should be obvious that this is NOT a power series and so does not have a "radius of convergence".

(The LaTeX codes are the same as before- it just has picked up a bug so you need to click on your "refresh" button everytime you use LaTeX.)
 
  • #3
HallsofIvy said:
If n is odd, [itex](-1)^n= -1[/itex] and if n is even, [itex](-1)^n= 1[/itex]. That means that if n is odd [itex](1+ (-1)^n)/2= 0[/itex] and if n is even, [itex](1+ (-1)^n)/2= 1[/itex]. Of course, if whether n is even or odd, n+ 1 is the opposite.

So for any odd n, [itex]5^{(1+(-1)^n)/2}= 5^0= 1[/itex] and [itex]x^{(1+ (-1)^{n+1})/2}= x^1= x[/itex] while for n even, [itex]5^{(1+(-1)^n)/2}= 5^1= 5[/itex] and [itex]x^{(1+ (-1)^{n+1})/2}= x^0= 1[/itex]. That is, [itex]5^{(1+(-1)^n)/2}x^{(1+(-1)^n)/2}[/itex] just alternates 5 and x as your sequence does.
I know that, no need to elaborate. I'm just wondering how can I come up with that equation if you only have the values "5, x, 10, x, ..." as a given.
HallsofIvy said:
As for finding the "radius of convergence", it should be obvious that this is NOT a power series and so does not have a "radius of convergence".

(The LaTeX codes are the same as before- it just has picked up a bug so you need to click on your "refresh" button everytime you use LaTeX.)
Uhm, I don't get it. Why is this not a power series?
 

1. What is summation notation?

Summation notation, also known as sigma notation, is a mathematical shorthand used to represent the sum of a series of numbers. It is written in the form of Σ (the capital Greek letter sigma) followed by the expression to be summed, a lower bound for the index variable, and an upper bound for the index variable. For example, Σk=1 to n k^2 represents the sum of the squares of all integers from 1 to n.

2. How do you find the summation notation for a given series?

To find the summation notation for a given series, you need to identify the pattern or rule that the series follows. This can be done by looking at the numbers in the series and determining the relationship between them. Once the pattern is identified, you can write the expression for the series using summation notation.

3. What is the radius of convergence?

The radius of convergence is a property of a power series, which is a series in the form of ∑a_n(x-c)^n. It represents the distance from the center point, c, to the nearest point where the series converges. In other words, it is the maximum value of x for which the series will converge.

4. How do you find the radius of convergence for a power series?

To find the radius of convergence for a power series, you can use the ratio test. The ratio test compares the absolute value of the ratio of consecutive terms in the series to a specific limit, and if the limit is less than 1, the series will converge. The radius of convergence is then equal to the reciprocal of this limit.

5. Why is it important to find the radius of convergence?

The radius of convergence is important because it tells us the range of values for which the power series will converge. It also helps us determine the behavior of the series at the endpoints of the radius. If the value of x is within the radius of convergence, the series will converge to a specific value. But if the value of x is outside the radius, the series will either diverge or converge to a different value. Therefore, knowing the radius of convergence is crucial in analyzing and using power series in various mathematical applications.

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