Solving the Limit of a[n] as n Goes to Infinity

In summary, the conversation discusses finding the limit of the sequence lim n->infinity of a[n] as x's absolute value is greater than 1. The solution involves using the ratio test to simplify the series and potentially using index shifts or differentiating a geometric series. It is also mentioned that the sum of a series can be found even if it is not a geometric series or a telescoping sum.
  • #1
flyerpower
46
0

Homework Statement


[PLAIN]http://admitere.ncit.pub.ro/moodle/filter/tex/pix.php/5c2cef253f2db3240db03f8c9b6c9463.gif

lim n->infinity of a[n] = ?

Homework Equations


|x| > 1

The Attempt at a Solution


Well, actually i figured out that the sequence converges, and I've tried to solve it using riemannian sums and integrals but no success.

I've also tried to solve the sum and then calculate the limit but i had no idea where to start from.

May anyone suggest me a starting point?
 
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  • #2
I think we can find the sum of a series only if it is a geometric series or a telescoping sum, and from a quick glance this doesn't look like either one.
Otherwise we would need some math software to approximate the sum.
 
  • #3
flyerpower said:

Homework Statement


[PLAIN]http://admitere.ncit.pub.ro/moodle/filter/tex/pix.php/5c2cef253f2db3240db03f8c9b6c9463.gif

lim n->infinity of a[n] = ?

Homework Equations


|x| > 1


The Attempt at a Solution


Well, actually i figured out that the sequence converges, and I've tried to solve it using riemannian sums and integrals but no success.

I've also tried to solve the sum and then calculate the limit but i had no idea where to start from.

May anyone suggest me a starting point?

Hint: think about what happens when you take derivatives of a geometric series.
 
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  • #4
flyerpower said:

Homework Statement


[PLAIN]http://admitere.ncit.pub.ro/moodle/filter/tex/pix.php/5c2cef253f2db3240db03f8c9b6c9463.gif

lim n->infinity of a[n] = ?

Homework Equations


|x| > 1

The Attempt at a Solution


Well, actually i figured out that the sequence converges, and I've tried to solve it using riemannian sums and integrals but no success.

I've also tried to solve the sum and then calculate the limit but i had no idea where to start from.

May anyone suggest me a starting point?

use the ratio test. ignore the summation symbol for now.

After using ratio test. see if you can separate the functions so you have "(function 1) + or - (function 2)"

if you can do that, then you could probably telescope to solve the sum. If not, then it could be a geometric series, which should be easier to test (if it is or not) after u get it to a simplified form.

If that doesn't work you should retry it but changing the index. i.e do an index shift, sub k=K+1 (them summation will start at 0), index will be K, not k-1. ratio test should be easier

(also, isn't this question supposed to be something like, for what values of x or k, is this covnergant?)
 
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  • #5
cloud360 said:
use the ratio test. ignore the summation symbol for now.

After using ratio test. see if you can separate the functions so you have "(function 1) + or - (function 2)"

if you can do that, then you could probably telescope to solve the sum. If not, then it could be a geometric series, which should be easier to test (if it is or not) after u get it to a simplified form.

If that doesn't work you should retry it but changing the index. i.e do an index shift, sub k=K+1 (them summation will start at 0), index will be K, not k-1. ratio test should be easier

(also, isn't this question supposed to be something like, for what values of x or k, is this covnergant?)
The OP has already determined that the series converges. He/she is now trying to figure out what the series converges to.
 
  • #6
My hint still stands. Figure out the sum of r^(k+1). Differentiate twice and put r=1/x.
 
  • #7
I figured out how to do it differntiating a geometric series after Dick gave me that hint but i hadn't time to do it yesterday, i will come back later with the solution.
 
  • #8
magicarpet512 said:
I think we can find the sum of a series only if it is a geometric series or a telescoping sum

No, that's not true. For example, the sum of 1/n! is e.
 

FAQ: Solving the Limit of a[n] as n Goes to Infinity

What is the definition of a limit as n goes to infinity?

A limit as n goes to infinity is the value that a function approaches as n gets larger and larger without bound.

How do you solve a limit as n goes to infinity?

To solve a limit as n goes to infinity, you can use techniques such as substitution, factoring, or algebraic manipulation to simplify the function. You can also use the properties of limits, such as the limit of a sum being the sum of the limits.

3. What does it mean if a limit as n goes to infinity does not exist?

If a limit as n goes to infinity does not exist, it means that the function does not approach a specific value as n gets larger and larger. This could be due to the function oscillating or having a vertical asymptote at infinity.

4. Can a limit as n goes to infinity be negative or imaginary?

Yes, a limit as n goes to infinity can be negative or imaginary. The limit of a function is determined by its behavior as n gets very large, not by the sign or type of number it approaches.

5. What is the difference between a one-sided limit and a two-sided limit as n goes to infinity?

A one-sided limit as n goes to infinity only considers the behavior of the function as n approaches infinity from one direction (either from the positive or negative side). A two-sided limit as n goes to infinity considers the behavior of the function from both directions and requires that the limit exists for both sides to be equal.

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