Convergence Tests Calculator

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Example

Created on 2024-06-20Asked by Lucas Jones (Solvelet student)

Apply the ratio test to determine the convergence of the series

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

Solution

Step 1: Identify the series

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

. Step 2: Apply the ratio test:

L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|.

Step 3: Compute the limit:

L = \lim_{n \to \infty} \left| \frac{\frac{(n+1)^2}{2^{n+1}}}{\frac{n^2}{2^n}} \right| = \lim_{n \to \infty} \left| \frac{(n+1)^2 \cdot 2^n}{n^2 \cdot 2^{n+1}} \right| = \lim_{n \to \infty} \left| \frac{(n+1)^2}{2n^2} \right|.

Step 4: Evaluate the limit:

L = \lim_{n \to \infty} \left| \frac{n^2 + 2n + 1}{2n^2} \right| = \frac{1}{2}.

Step 5: Analyze the result:

L < 1 \implies \text{convergent}.

Step 6: Conclusion. By the ratio test, the series

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

converges. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Victoria Hill on Solvelet

1. Use the integral test to determine the convergence of the series

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

2. z

DefinitionConvergence tests:- A test or method to find which series converges and diverges. These test include under which conditions a series will be convergent absolutely, convergent conditionally, or divergent. Examples:- Ratio test, root test, comparison test, integral test, and alternating series test, these converge tests also tell us about the series whether it is convergent or divergent, or not the sequence of real numbers. These test methods are used in real and complex analysis, calculus and numerical analysis to estimate the value of a limit of a sequence of real number and checked which the sequence some of the terms such as limit, n ε IN, may converge, i.e., limit may exists or does not. Example:- Test of the ratio says that if the limit of the absolute value of the ratio of two terms of a series, as n approaches to infinity, is less than one, the series will be convergent absolutely; if it is greater than one, the series will be diverges; if it is equal or greater than one the test of the ratio is inconclusive. Hence, it can be mathematically expressed as.

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