Calculus 2 - Geometric Series, P-Series, Ratio Test, Root Test, Alternating Series, Integral Test

This paragraph introduces a variety of mathematical tests used to determine whether a given series will converge or diverge. It begins with the Divergence Test, which involves taking the limit of the sequence's term as n approaches infinity; if the limit is non-zero, the series diverges. The paragraph continues with the Geometric Series Test, emphasizing the importance of identifying the common ratio 'r' and its absolute value; if less than one, the series converges, otherwise, it diverges. The p-Series Test is also discussed, highlighting that for the series in the form of 1/n^p to converge, 'p' must be greater than one. The Telescoping Series Test is then introduced, explaining how to identify and sum series with canceling terms. Lastly, the Integral Test is briefly mentioned, suggesting that if the integral from one to infinity of a positive, continuous, and decreasing function 'f(x)' has a finite value, the series converges.

In this paragraph, the discussion on series convergence tests continues with a detailed explanation of the Integral Test. It explains that for a series with a positive, continuous, and decreasing function 'f(n)', if the integral from one to infinity of 'f(x)dx' yields a finite value, the series converges; otherwise, it diverges. The Ratio Test is then described, stating that the limit of the absolute value of the ratio of consecutive terms (a_(n+1)/a_n) as n approaches infinity, if less than one, indicates convergence, while greater than one or infinity indicates divergence. The Root Test is also outlined, where the limit of the nth root of the absolute value of the term (a_n) as n approaches infinity, if less than one, results in convergence, and greater than one or infinity in divergence. The Direct Comparison Test and Limit Comparison Test are introduced, explaining how the convergence or divergence of one series can be used to infer the behavior of another series. The paragraph concludes with the Alternating Series Test, which requires two conditions: passing the divergence test with a limit of zero and the sequence being decreasing.

This paragraph applies the previously discussed convergence tests to specific series problems. It starts with a series involving '2n^2 + 5' over '7n^2 - 4', suggesting the use of the Divergence Test and L'Hôpital's Rule to determine divergence. The paragraph then examines a series with 'cube root of n' divided by 'n to the fifth power', identifying it as a p-series and concluding convergence due to the exponent being greater than one. A geometric series example with 'five times one fourth raised to the power of n minus one' is provided, with the sum calculated using the formula for geometric series when the common ratio's absolute value is less than one. The paragraph also discusses an alternating series with 'negative 1 to the power of n' divided by the square root of 'n', confirming convergence using the Alternating Series Test and determining conditional convergence by examining the absolute value of the series.

The paragraph delves deeper into series convergence with examples and tests. It begins with a series '1/n * (n + 1)', suggesting partial fraction decomposition to identify it as a telescoping series, which converges to a finite value. Another series '1/n^2 + 4' is discussed, recommending the Direct Comparison Test to establish convergence by comparing it to a known p-series. The paragraph then considers a series '1/sqrt(n - 2)', using the Divergence Test and Direct Comparison Test to conclude divergence. The Integral Test is also applied to this series, confirming divergence due to the integral's result of infinity. The paragraph continues with a series 'sqrt(n) / (n^3 + 2)', recommending the Limit Comparison Test, which shows convergence by comparing it to a simplified p-series. The Root Test is applied to '3n^2 - 9 / (7n^2 + 4)^n', leading to the conclusion of convergence as the limit is less than one.

In this paragraph, complex series are analyzed using various convergence tests. A series '2^n / n!' is introduced, and the Ratio Test is applied to determine its convergence. The limit of the ratio of consecutive terms, when simplified, shows that the series converges as the limit approaches zero. This final example wraps up the discussion by demonstrating the application of the Ratio Test to a series with a factorial in the denominator, a common scenario in convergence analysis.