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Example:
Does the sequence an=1/n converge or diverge?
SOLUTION: limn→∞ 1/n=0; hence the sequence converges to 0.
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If an is a sequence of real numbers, then an infinite series is an expression of the form
The elements in the sum are called terms; an is the nth or general term of the series.
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Example :
The p-series with p = 1 is called the harmonic series:
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THEOREM 2a
THEOREM 2b
(where m is any positive integer) both converge or both diverge. (Note that the sums most likely will differ.)
THEOREM 2c
both converge or both diverge. (Again, the sums will usually differ.)
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Example:
SOLUTION:
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THE GEOMETRIC SERIES TEST
Example :
Does 0.3 + 0.03 + 0.003 + . . . converge or diverge?
SOLUTION:
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The series ∑an is called a nonnegative series if an ≥ 0 for all n.
Example:
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SOLUTION: The associated improper integral is
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(1)If ∑un converges and an ≦ un, then ∑an converges.
(2)If ∑un diverges and an ≧ un, then ∑an diverges.
Any known series can be used for comparison. Particularly useful are p-series, which converge if p > 1 but diverge if p ≦ 1, and geometric series, which converge if |r| < 1 but diverge if |r| ≧ 1.
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Example :
Does ∑1/1+n4 converge or diverge?
Solution:
Any test that can be applied to a nonnegative series can be used for a series all of whose terms are negative. We consider here only one type of series with mixed signs, the so-called alternating series. This has the form:
where ak > 0.
The series
is the alternating harmonic series.
(1)an + 1 < an for all n
(2)limn→∞an=0
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Example:
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Example:
Solution:
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An expression of the form
where the as are constants, is called a power series in x; and
is called a power series in (x − a).
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Example :
Find all x for which the following series converges:
Solution:
hus, the radius of convergence is 1. The endpoints must be tested separately since the Ratio Test fails when the limit equals 1. When x = 1, (3) becomes 1 + 1 + 1 + . . . and diverges; when x = −1, (3) becomes 1 − 1 + 1 − 1 + . . . and diverges. Thus the interval of convergence is −1 < x < 1.
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Let the function f be defined by
its domain is the interval of convergence of the series.
Functions defined by power series behave very much like polynomials, as indicated by the following properties:
Note that power series (1) and its derived series (2) have the same radius of convergence but not necessarily the same interval of convergence.
Example:
Find the intervals of convergence of the power series for f(x) and f′(x).
SOLUTION:
PROPERTY 2C. The series obtained by integrating the terms of the given series (1) converges to ∫xaf(t) dt for each x within the interval of convergence of (1); that is,
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c0 + c1 (x − a) + c2 (x − a)2 + . . . + cn(x − a)n + . . .
on an interval |x − a| < r, then the coefficients are given by
The series
is called the Taylor series of the function f about the number a. There is never more than one power series in (x − a) for f(x). It is required that the function and all its derivatives exist at x = a if the function f(x) is to generate a Taylor series expansion.
When a = 0 we have the special series
called the Maclaurin series of the function f; this is the expansion of f about x = 0.
Example:
Find the Maclaurin series for f(x) = ex.
SOLUTION:
The function f(x) at the point x = a is approximated by a Taylor polynomial Pn(x) of order n:
The Taylor polynomial Pn(x) and its first n derivatives all agree at a with f and its first n derivatives. The order of a Taylor polynomial is the order of the highest derivative, which is also the polynomial’s last term.
In the special case where a = 0, the Maclaurin polynomial of order n that approximates f(x) is
The Taylor polynomial P1(x) at x = 0 is the tangent-line approximation to f(x) near zero given by
Example :
Find the Taylor polynomial of order 4 at 0 for f(x) = e−x. Use this to approximate f(0.25).
SOLUTIONS:
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When we approximate a function using a Taylor polynomial, it is important to know how large the remainder (error) may be.
TAYLOR’S THEOREM
If a function f and its first (n + 1) derivatives are continuous on the interval |x − a| < r, then for each x in this interval
where
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and c is some number between a and x. Rn(x) is called the Lagrange remainder.
Note that the equation above expresses f(x) as the sum of the Taylor polynomial Pn(x) and the error that results when that polynomial is used as an approximation for f(x).
When we truncate a series after the (n + 1)st term, we can compute the error bound Rn, according to Lagrange, if we know what to substitute for c. In practice we find, not Rn exactly, but only an upper bound for it by assigning to c the value between a and x that determines the largest possible value of Rn. Hence:
Example :
Estimate the error in using the Maclaurin series generated by e x to approximate the value of e.
SOLUTION:
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C6. Computations with Power Series
Example:
Compute 1/√e to four decimal places.
SOLUTION:
We can use the Maclaurin series,
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A complex number is one of the form a + bi, where a and b are real and i 2 = −1. If we allow complex numbers as replacements for x in power series, we obtain some interesting results.
Consider, for instance, the series
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