Sequences We investigate sequences. Representing sequences visually We can graph the terms of a sequence and find functions of a real variable that coincide with sequences on their common domains. Limits of sequences There are two ways to establish whether a sequence has a limit. What is a series A series is an infinite sum of the terms of sequence. Special Series We discuss convergence results for geometric series and telescoping series. The divergence test.
The divergence test If an infinite sum converges, then its terms must tend to zero. The Integral test. The integral test Certain infinite series can be studied using improper integrals. The alternating series test Alternating series are series whose terms alternate in sign between positive and negative. There is a powerful convergence test for alternating series. Dig-In: Estimating Series We learn how to estimate the value of a series. Remainders for Geometric and Telescoping Series For a convergent geometric series or telescoping series, we can find the exact error made when approximating the infinite series using the sequence of partial sums.
Remainders for alternating series There is a nice result for approximating the remainder of convergent alternating series. Remainders and the Integral Test There is a nice result for approximating the remainder for series that converge by the integral test.
The ratio test Some infinite series can be compared to geometric series. The root test Some infinite series can be compared to geometric series. The comparison test We compare infinite series to each other using inequalities. The limit comparison test We compare infinite series to each other using limits. Absolute and Conditional Convergence. Absolute and Conditional Convergence The basic question we wish to answer about a series is whether or not the series converges.
If a series has both positive and negative terms, we can refine this question and ask whether or not the series converges when all terms are replaced by their absolute values. This is the distinction between absolute and conditional convergence, which we explore in this section. Approximating functions with polynomials. Higher Order Polynomial Approximations We can approximate sufficiently differentiable functions by polynomials.
Power series Infinite series can represent functions. Introduction to Taylor series. Introduction to Taylor series We study Taylor and Maclaurin series. Numbers and Taylor series. Numbers and Taylor series Taylor series are a computational tool. Calculus and Taylor series. Calculus and Taylor series Power series interact nicely with other calculus concepts. Differential equations Differential equations show you relationships between rates of functions.
Separable differential equations. Separable differential equations Separable differential equations are those in which the dependent and independent variables can be separated on opposite sides of the equation. Parametric equations We discuss the basics of parametric curves. Calculus and parametric curves We discuss derivatives of parametrically defined curves.
Introduction to polar coordinates. Introduction to polar coordinates Polar coordinates are coordinates based on an angle and a radius. Gallery of polar curves We see a collection of polar curves. Derivatives of polar functions.
Derivatives of polar functions We differentiate polar functions. Integrals of polar functions. Integrals of polar functions We integrate polar functions. Working in two and three dimensions. For which values of is guaranteed to converge? For which values of if any, does converge? Note that the ratio and root tests are inconclusive. Using the hint, there are terms for and for each term is at least Thus, which converges by the ratio test for For the series diverges by the divergence test.
Suppose that for all Can you conclude that converges? Let where is the greatest integer less than or equal to Determine whether converges and justify your answer. One has The ratio test does not apply because if is even. However, so the series converges according to the previous exercise. Of course, the series is just a duplicated geometric series.
The following advanced exercises use a generalized ratio test to determine convergence of some series that arise in particular applications when tests in this chapter, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if then converges, while if then diverges. Let Explain why the ratio test cannot determine convergence of Use the fact that is increasing to estimate. Let Show that For which does the generalized ratio test imply convergence of Hint: Write as a product of factors each smaller than.
The inverse of the factor is so the product is less than Thus for The series converges for. Let Show that as. True or False? Justify your answer with a proof or a counterexample. If then converges. If then diverges. If converges, then converges. Is the sequence bounded, monotone, and convergent or divergent? If it is convergent, find the limit. Is the series convergent or divergent? If convergent, is it absolutely convergent? A legend from India tells that a mathematician invented chess for a king.
The king enjoyed the game so much he allowed the mathematician to demand any payment. The mathematician asked for one grain of rice for the first square on the chessboard, two grains of rice for the second square on the chessboard, and so on. Find an exact expression for the total payment in grains of rice requested by the mathematician. Assuming there are grains of rice in pound, and pounds in ton, how many tons of rice did the mathematician attempt to receive?
The following problems consider a simple population model of the housefly, which can be exhibited by the recursive formula where is the population of houseflies at generation and is the average number of offspring per housefly who survive to the next generation.
Assume a starting population. Find if and. Find an expression for in terms of and What does it physically represent? If and find and. For what values of will the series converge and diverge? What does the series converge to? Skip to content Sequences and Series. Learning Objectives Use the ratio test to determine absolute convergence of a series. Use the root test to determine absolute convergence of a series. Describe a strategy for testing the convergence of a given series.
Ratio Test Consider a series From our earlier discussion and examples, we know that is not a sufficient condition for the series to converge. Ratio Test. Proof Let be a series with nonzero terms. Using the Ratio Test. From the ratio test, we can see that. The series converges. Root Test The approach of the root test is similar to that of the ratio test.
Using the Root Test. To apply the root test, we compute. Choosing a Convergence Test At this point, we have a long list of convergence tests. Using Convergence Tests. The series is not a or geometric series. The series is not alternating.
For large values of we approximate the series by the expression. Series Converging to and. Key Concepts For the ratio test, we consider. Chapter Review Exercises True or False? Glossary ratio test for a series with nonzero terms, let if the series converges absolutely; if the series diverges; if the test is inconclusive root test for a series let if the series converges absolutely; if the series diverges; if the test is inconclusive.
Previous: Alternating Series. Next: Introduction. Share This Book Share on Twitter. Divergence Test For any series evaluate. If the test is inconclusive. Geometric Series. If the series converges to. Any geometric series can be reindexed to be written in the form where is the initial term and is the ratio. If the series converges. Sign up to join this community. The best answers are voted up and rise to the top.
Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. How to show convergence or divergence of a series when the ratio test is inconclusive?
Ask Question. Asked 8 years, 4 months ago. Active 8 years, 4 months ago. Viewed 3k times. So how can I approach this? Or was I on the right track and did something wrong? Mirrana Mirrana 8, 33 33 gold badges 77 77 silver badges bronze badges.
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