Ultimate calculus 2 study

test for divergence

test for divergence

A test used to determine the convergence or divergence of a series. If the limit of the terms of a series is not zero, then the series diverges. If the limit is zero or does not exist, further tests are needed to determine convergence.

calculus integral test

The integral test is a method in calculus used to determine the convergence or divergence of an infinite series by comparing it to the convergence or divergence of an improper integral. It states that if the integral of the series terms converges, then the series itself also converges. Conversely, if the integral diverges, then the series also diverges. This test is particularly useful when dealing with series that have non-negative terms.

the comparison test calculus

The comparison test is a method used in calculus to determine the convergence or divergence of an infinite series. It states that if the terms of a series are nonnegative and can be compared to the terms of a known convergent series, then the original series will also converge. Similarly, if the terms of a series can be compared to the terms of a known divergent series, then the original series will also diverge.

the limit comparison test

The limit comparison test is a method used in calculus to determine the convergence or divergence of a series. It states that if the limit of the ratio of the terms of a given series to the terms of a known convergent series is a finite positive number, then both series either converge or diverge together. If the limit is zero or infinity, the test is inconclusive.

the alternating series test

The alternating series test is a method used to determine the convergence or divergence of an alternating series. It states that if the terms of an alternating series decrease in absolute value and approach zero, then the series is convergent. Additionally, the absolute value of each term must be greater than the absolute value of the subsequent term. If these conditions are met, the series is convergent. If not, the series is divergent.

ratio test calculus

The ratio test is a convergence test used in calculus to determine the convergence or divergence of a series. It is based on the comparison of the ratio of consecutive terms in the series. If the limit of this ratio as n approaches infinity is less than 1, the series converges. If the limit is greater than 1 or does not exist, the series diverges. If the limit is exactly 1, the test is inconclusive and another test may be needed.

root test calculues

The root test is a convergence test used in calculus to determine the convergence or divergence of a series. It states that if the limit of the absolute value of the nth root of the terms of a series is less than 1, then the series converges. If the limit is greater than 1 or undefined, then the series diverges.

example using root test

The root test is a convergence test used in calculus to determine the convergence or divergence of a series. It is based on the comparison of the nth root of the absolute value of the terms of the series with 1.

If the limit of the nth root of the absolute value of the terms of the series is less than 1, the series converges. If the limit is greater than 1 or undefined, the series diverges. If the limit is exactly 1, the test is inconclusive.

To illustrate the root test, let's consider an example:

Suppose we have the series ∑(n=1 to infinity) (n^2)/(2^n).

To apply the root test, we take the nth root of the absolute value of the terms:

lim(n→∞) [(n^2)/(2^n)]^(1/n)

Simplifying this expression, we get:

lim(n→∞) (n^(2/n))/(2)

Taking the limit as n approaches infinity, we find:

lim(n→∞) (n^(2/n))/(2) = 1/2

Since the limit is less than 1, the series converges by the root test.

example using ratio test

The ratio test is used to determine the convergence or divergence of a series. It states that if the limit of the absolute value of the ratio of consecutive terms in a series is less than 1, then the series converges. If the limit is greater than 1 or does not exist, the series diverges. An example using the ratio test is:

Consider the series:

[ \sum_{n=1}^{\infty} \frac{n^2}{2^n} ]

To apply the ratio test, we take the limit of the absolute value of the ratio of consecutive terms:

[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{\frac{{(n+1)}^2}{2^{n+1}}}{\frac{n^2}{2^n}} \right| ]

Simplifying, we get:

[ \lim_{n \to \infty} \left| \frac{{(n+1)}^2}{2n^2} \right| = \lim_{n \to \infty} \left| \frac{n^2 + 2n + 1}{2n^2} \right| = \frac{1}{2} ]

Since the limit is less than 1, the series converges by the ratio test.