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Two related tests for checking series convergence

The ratio test and the root test are two more ways of checking for convergence of infinite series. We will show near the end of the section that the two are actually related. The ratio test asks whether, in the limit that the number of terms goes to infinity (n → ∞), the ratio of the (n+1)th term to the nth term is less than one.

The root test checks whether the nth root of the nth term is less than one. We'll see why that implies series convergence in the proof.

The ratio test

Proof: To prove the ratio test, we compare the result of our limit, in generic terms, to the convergent geometric series with terms rn, where r < 1.

The integral is a simple one (I've flipped the exponent and denominator of the integral from -p+1 to 1-p for convenience).

R→∞. With the p-integral in hand, we can use the integral test to determine which p-series converge.

Example 1:   Ratio test, convergent series

Solution: We'll set up the ratio test for this series and see how it goes. In setting up these ratios remember that n has to become n+1 in the numerator:

Now generally we'll get a single fraction by recognizing that division is multiplication by the reciprocal:

Now we can use the properties of factorials and the laws of exponents to do some simplification. Recall that 2n+1 = 2n ยท 21 and (n + 2)! = (n + 2)(n + 1)!, so we have:

which reduces to

The limit is less than one, so the series converges. Hopefully, you had a pretty good suspicion that this series was convergent from the beginning, because the factorial function grows much more rapidly than an exponential function for n > N, some number at or beyond the point where the two functions cross.

Remember that this test, and any convergence test doesn't necessarily tell us which number the series converges to, just that it does. We would need to find the sum of the series in another way.

Example 2:   Ratio test, divergent series

Solution: First set up the ratio test:

Now, recognizing that en+1/en   =   ene1/en   =   e, and using L'Hopital's rule to evaluate the limit, we get:

The series diverges. Notice that this series wouldn't have passed the divergence test, either.

Example 3:   Ratio test, inconclusive

Solution: We begin by setting up the ratio-test expression. I'm going to forgo the absolute value signs this time (just assume they're there), and I'll just write the denominator as a reciprocal product right away:

Now expand the (n+1)2 and rearrange to keep the roots together:

The properties of limits allow us to separate this expression into a product of limits:

The properties of limits also allow us to put the power of 1/2 on the left outside of the limit ...

... to obtain a limit of 1, which means that the ratio test is inconclusive for this series.

This series can be shown to be convergent by direct comparison to the p-series with p = 3/2.

The root test

Proof: To prove the root test we ...

Example 4:   Root test

Solution: This series is a perfect candidate for the root test because taking the nth root will vastly simplify it. Here goes:

Now we can use L'Hopital's rule to find the limit:

... so the series converges.

Example 5:   Root test

Solution: Here is another great candidate for the root test because of the powers of n. Set up the limit like this:

Because 0 < 1, the series converges. Notice that the series with terms -3/n would not converge (-3 times an arithmetic series, but that's not what's going on here).

Example 6:   Root test

Solution: First set up the limit. The nth root gives us 2/3.

Now the n3/n can be rewritten as (n1/n)3, which is one as n → ∞:

... so the series converges.

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