SERIES

Definition. Let {a_n} be a sequence. The n^th partial sum, s_n, is the sum

s_n = a₁ + a₂ + ... + a_n

We obtain the sequence of partial sums {s_n}.

Definition. Given a sequence {a_n} and the sequence of its partial sums s_n, then we say that the series

is convergent if the sequence s_n is convergent and has finite limit. If

then we write

If the sequence s_n is not convergent then we say that the series

is divergent.

Convergence of geometric series.

Example.

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Divergence of harmonic series.

Theorem. n^th Term Test. If the series

is convergent, then

Equivalently, if

then the series is divergent.

Examples.

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Telescoping series.

Examples.

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INTEGRAL TEST

Theorem. The Integral Test. Let

be a series such that there exists a function f with the following properties:

1. a_n = f(n)
2. f is continuous, positive and decreasing.

Then

• If the integral

converges, then the series

converges.

• If the integral

diverges, then the series

diverges.

Examples.

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Theorem. p-Series Test. The p-series

is convergent if p > 1 and is divergent if p < 1.

INTEGRAL TEST ESTIMATE

Theorem. The Integral Test Estimate. Suppose that

is a series which satisfies the hypotheses of the Integral Test using the function f and which converges to L. Let s_n = a₁ + a₂ + ... + a_n be the n^th partial sum and let r_n = L - s_n. Then

• Problem. Find an approximation of the series

using the partial sum s₂₀. What is the maximum possible error using this approximation?

• Problem. Find an approximation of the series

using the partial sum s₃₀. What is the maximum possible error using this approximation?

• Problem. Find an integer n such that, using s_n as an approximation of the series,

the maximum possible error is at most .00001.

If we add s_n to each term of the error estimate given in the theorem above, we obtain the following which provides a way to obtain a better estimate for value of the power series.

Corollary.

• Problem. Using the Corollary, find an approximation for the series

with n = 100 and find the maximum possible error in using this approximation.

COMPARISON TEST

Theorem. The Comparison Test. Suppose that

and

are series such that 0 < a_n < b_n for all n.

1. If is convergent then is also convergent.
2. If is divergent then is also divergent.

Exercises:

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LIMIT COMPARISON TEST

Theorem. The Limit Comparison Test. Suppose that

and

are series such that a_n and b_n are positive for all n.

1. If then and are either both convergent or both divergent.
2. If then convergent implies that is also convergent.
3. If then divergent implies that is also divergent.

Exercises:

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ALTERNATING SERIES

Theorem. The Alternating Series Test. Suppose that {a_i} is a sequence of positive numbers such that

1. a_i > a_i+1 for all i.
2. 

Then the series is convergent.

Exercises:

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Theorem. The Alternating Series Error Estimate. Suppose that {a_i} is a sequence of positive numbers which satsifies the hypothesis of the theorem above. Suppose that the series

converges to L; let r_n = L - s_n where s_n denotes the n^th partial sum. Then |r_n| < a_n+1.

• Problem. Find an approximation of the series

using the partial sum s₁₀₀. What is the maximum possible error using this approximation?

• Problem. Find an integer n such that, using s_n as an approximation of the series

the maximum possible error is at most .0001.

RATIO TEST

Theorem. The Ratio Test. Suppose that {a_i} is a sequence of positive numbers

1. If then is convergent.
2. If then is divergent.

Exercises:

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Attention. The ratio test gives no information when

Exercises:

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ABSOLUTE CONVERGENCE

Definition. A series is absolutely convergent if the series is convergent.

A series is conditionally convergent if the series is convergent but is not absolutely convergent.

Example.

• is conditionally convergent.

Theorem. If a series is absolutely convergent then it is convergent.

Exercises:

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Corollary. The Absolute Ratio Test.

1. If then is convergent.
2. If then is divergent.

Attention. The ratio test gives no information when