Lab Activity #12

Math 1215 Power, Maclaurin,

Calculus for Engineers II and Taylor Series

Work with your group to solve the following problems; write each problem’s solution
clearly separated from solutions of other problems. Before leaving the lab, you must
have all of your solutions checked by the GTA.
1
1. Express the function f (x) = as a power series centered at a = 0. What is its interval
1 + x7
of convergence?

2. Using the definition of Macalurin series, find the Maclaurin series of the function

f (x) = ln(1 + x).

3. Use known Maclaurin series to obtain the Maclaurin series for each of the following functions.
For each Maclaurin series, find the interval of convergence.
(a) sin2 (x) (Hint: Differentiate and use 2 sin(x) cos(x) = sin(2x))
 2
x
(b) arctan
9

∞
X (2x + 3)k
k
4. Consider the power series (−1) .
k=2
k ln(k)
(a) What is the center a of the power series?
(b) What are the coefficients ck of the power series?
(c) What is the radius of convergence of the power series?
(d) What is the interval of convergence of the power series?

5. Use known Maclaurin series to evaluate each of the following series.

∞
X (−3)k
(a)
k=1
5k k!
∞
X (−1)n π 2n
(b) (Hint: 36 = 62 )
n=1
36n (2n)!
∞
X (−1)j
(c) √
j=2
( 3)2j+1 (2j + 1)
Tests for Series
∞
X
(1) Test for Divergence: Used only to show divergence of series of the form ak ; the test NEVER implies
k=0
∞
X
convergence of a series. The Test for Divergence states that ak necessarily diverges if lim ak ̸= 0.
k→∞
k=0
∞
X
(2) Geometric Series: A geometric series is any series of the form a·rk , identified by the ratio of consecutive
k=0
ak+1
terms being constant; that is, = r for k = 0, 1, 2, 3, . . ..
ak
(3) Integral Test: Suppose f is satisfies the following properties: (i) f is continuous, (ii) f is positive, and (iii)
f is decreasing for x ≥ 1. Let ak = f (k) for k = 1, 2, . . .. Then
∞
X Z ∞
ak and f (x) dx
k=1 1

either both converge or both diverge. In the case of convergence, the value of the integral is not necessarily
equal to the value of the series.
∞
X 1
(4) p-series Test: The p-series converges for p > 1 and diverges for p ≤ 1.
kp
k=1
∞
X an+1 an+1
(5) Ratio Test: The series an is absolutely convergent if lim < 1 and diverges if lim > 1.
n=1
n→∞ an n→∞ an
If the limit equals 1, the test is inconclusive.
∞
X p p
n n
(6) Root Test: The series an is absolutely convergent if lim |an | < 1 and diverges if lim |an | > 1. If
n→∞ n→∞
n=1
the limit equals 1, the test is inconclusive.
∞
X
(7) Comparison Test: Given an with an ≥ 0, the series converges if there exists 0 ≤ bn such that an ≤ bn
n=1
∞
X
and bn converges (you must supply the bn ); the given series diverges if there exists 0 ≤ cn such that
n=1 P∞
cn ≤ an and n=1 cn diverges (you must supply the cn ).
∞
X an
(8) Limit Comparison Test: Given an , suppose there exists bn such that 0 ≤ lim is nonnegative and
n=1
n→∞ bn
∞
X ∞
X
finite. If bn converges, then the given series converges; similarly, if bn diverges, then the given series
n=1 n=1
also diverges.
P∞
(9) Alternating Series Test: Given n=1 (−1)n bn where bn > 0, the series converges if bn is decreasing and
lim bn = 0. The remainder RN for an alternating series satisfies RN < bN +1 (i.e., the remainder is less than
n→∞
the first term left off).
∞
X ∞
X
(10) Absolute Convergence: The series an is absolutely convergent if |an | converges.
n=1 n=1
∞
X P∞
(11) Conditional Convergence: The series an is conditionally convergent if n=1 an converges but is NOT
n=1
absolutely convergent.
 Taylor/Maclaurin Polynomials and Series
D.C

Commonly Used Taylor Series

1. Geometric
P∞ Series :
1 n
1−x
= n=0 x for x ∈ (−1, 1)

2. Exponential
P∞ xn Function:
x
e = n=0 n! for x ∈ R

3. Cosine P
functions:
(−1)n x2n
cos x = ∞n=0 (2n)!
for x ∈ R

4. Sine function:
(−1)n x2n+1
sin x = ∞
P
n=0 (2n+1)! for x ∈ R
P∞ (−1)n+1 xn
5. Natural Logarithm:ln(1 + x) = n=1 n
for x ∈ (−1, 1]
P∞ (−1)n x2n+1
6. Inverse Tangent:arctan(x) = n=0 2n+1
for x ∈ [−1, 1]

Taylor and Maclaurin Polynomials

Let f : I → R be a function whose derivatives f (n) exist on an interval I, and let x0 ∈ I.
Definition: Taylor Polynomial (Order N )
N
X f (n) (x0 )
pN (x) = (x − x0 )n
n=0
n!
If x0 = 0, this becomes the Maclaurin polynomial:
N
X f (n) (0)
pN (x) = xn
n=0
n!

Definition: Taylor Series

∞
X f (n) (x0 )
P∞ (x) = (x − x0 )n
n=0
n!
This converges to f (x) if limN →∞ RN (x) = 0, where

RN (x) = f (x) − pN (x)