Calculus 2 (Power Series) when the limit is zero by root test

In summary, when solving power series problems using the root test, we should take the absolute value of the variable and use the fact that absolute convergence implies convergence. We can also consider the radius of convergence to be infinite in this case. The steps to solve such a problem would include taking the absolute value, using the root test, and considering the radius of convergence.
  • #1
yeny
7
0
Hi guys!

Here's a problem i was working on. I solved it by root test and got absolute value of x on the outside of the limit and the limit equaled zero. Is it wrong to multiply the outside absolute value by the zero I got from the limit? or is that okay?

In general, when we are solving power series problems, is it okay to think of R equals infinity when the limit is zero? is that always the case? the interval of convergence is (-inf, +inf)

what are the steps that YOU would take to solve such a problem?

hope this makes sense. THANK YOU !

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  • #2
yeny said:
Hi guys!

Here's a problem i was working on. I solved it by root test and got absolute value of x on the outside of the limit and the limit equaled zero. Is it wrong to multiply the outside absolute value by the zero I got from the limit? or is that okay?

In general, when we are solving power series problems, is it okay to think of R equals infinity when the limit is zero? is that always the case? the interval of convergence is (-inf, +inf)

what are the steps that YOU would take to solve such a problem?

hope this makes sense. THANK YOU !
The root test only works for series of positive terms. So you should start by taking the absolute value of $|x|$. When you find that the series converges you can then use the fact that absolute convergence implies convergence to deduce the result for negative $x$.

Apart from that, your answer is correct. The limit of the $k$th root is zero, so you can conclude that the series converges (absolutely) for all $x$. That is expressed informally by saying that the radius of convergence is infinite.
 

1. What is the root test in Calculus 2?

The root test is a method used to determine the convergence or divergence of a series. It involves taking the nth root of the absolute value of each term in the series and then finding the limit as n approaches infinity. If the limit is less than 1, the series converges. If the limit is greater than 1, the series diverges. If the limit is equal to 1, the test is inconclusive and another method must be used.

2. How is the root test used in Calculus 2?

In Calculus 2, the root test is used to determine the convergence or divergence of a power series. By taking the nth root of the coefficients of the series and finding the limit as n approaches infinity, we can determine if the series converges or diverges. If the limit is less than 1, the series converges. If the limit is greater than 1, the series diverges. If the limit is equal to 1, the test is inconclusive and another method must be used.

3. What is the difference between a convergent and divergent series?

A convergent series is one in which the sum of the terms approaches a finite value as the number of terms increases. In other words, the series "converges" to a specific value. On the other hand, a divergent series is one in which the sum of the terms either increases without bound or oscillates between different values, and therefore does not have a finite sum.

4. Can the root test be used on all series in Calculus 2?

No, the root test can only be used on series with positive terms. If the terms of a series are negative, the test cannot be applied. In addition, the root test is not always conclusive and may require the use of other tests or methods to determine the convergence or divergence of a series.

5. Why is it important to determine the convergence or divergence of a series?

Determining the convergence or divergence of a series is important because it allows us to understand the behavior of the series and make predictions about its sum. In addition, many real-world applications in fields such as physics, engineering, and economics rely on the convergence or divergence of series to make accurate calculations and predictions.

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