CALCULUS 3
·
INFINITE SERIES
The Integral
Comparison and Limit
Comparison
How do we know if an infinite sum converges?
Group 2: Atillo · Dela Cruz · Gahi · Macasa · Olarte
∑
∫
Learning Objectives
By the end of the 90-minute session, you will be able to:
Cognitive
Affective
Psychomotor
Classify series as convergent
or divergent using the
Integral Test and the Limit
Comparison Test.
Appreciate why a test’s
conditions must be checked
before concluding.
Construct and graph y = 1/x
and y = 1/x² with rectangles
in Desmos, and complete
the partial-sum table.
Target: at least 4 of 6 series in
Your Turn
Target: 1 idea in class + exit ticket
Target: at least 10 of 12 entries
correct
The Never-Ending Snack
Eat half a pizza, then half of what’s left, then half again… forever.
½
¼
⅛
total → 1 whole pizza
Day n: scroll for 1/n hours
1+½+⅓+¼+…
Must an infinite sum of positive
numbers always be infinite?
Quick Recall
1
Series
∑ aₙ converges if the partial sums Sₙ
approach a finite number
3
Improper integral
∫ 1/x² dx from 1 to ∞ = 1
2
Geometric
∑ rⁿ converges when |r| < 1
4
Limits at ∞
lim (3n²+1)/(n²+5) = 3 (leading terms
decide)
Activity 1: Do the sums level off?
Partial sums Sₙ of two series
3,5
∑ 1/n²
3
levels off near 1.6
bounded above
2,5
2
1,5
1
∑ 1/n
0,5
0
1
2
3
4
5
6
N
∑ 1/n
∑ 1/n²
7
8
9
10
keeps growing
no upper limit
Activity 1: Areas and Sums
In pairs, using Desmos
1
Graph y = 1/x and y = 1/x² for x ≥ 1
2
On y = 1/x², draw rectangles of
height 1/n² on [n−1, n]
3
On y = 1/x, draw rectangles of
height 1/n on [n, n+1]
4
Compare rectangle area with area
under the curve
Rectangles under 1/x² → total stays below 1 + 1 = 2. Rectangles above 1/x → total ≥ ln(N+1) → ∞.
The Integral Test
Compare a series with an improper integral
∑ aₙ and ∫ f(x) dx both converge or both diverge
where aₙ = f(n) and f is…
Positive
Continuous
Decreasing
f(x) > 0
no breaks on [k, ∞)
f′(x) ≤ 0
The test tells us whether a series converges, not what it adds up to.
Using the Integral Test
1
2
3
4
Define f(x) from aₙ
Check the 3
conditions
Evaluate ∫ f(x) dx
State the conclusion
p-series shortcut
∑ 1/nᵖ
p>1
converges
p≤1
diverges
Example: Integral Test
Does ∑ n·e⁻ⁿ² converge?
1
f(x) = x·e⁻ˣ² is positive and continuous on [1, ∞)
2
f′(x) = e⁻ˣ²(1 − 2x²) < 0 for x ≥ 1, so f is decreasing
3
∫ x·e⁻ˣ² dx from 1 to ∞ = 1/(2e) (a finite number)
The integral converges, so the series CONVERGES
Limit Comparison Test
Compare two series through a limit
(2n+1)/(n³+n+4) ≈ 2n/n³ = 2/n² for large n
Let aₙ > 0, bₙ > 0 and L = lim aₙ/bₙ
0<L<∞
L=0
L=∞
Same behavior as ∑ bₙ
∑ bₙ converges → ∑ aₙ
converges
∑ bₙ diverges → ∑ aₙ diverges
Watch out: if L = 0 but ∑ bₙ diverges, the test is inconclusive.
Example: Limit Comparison Test
Does ∑ (2n+1)/(n³+n+4) converge?
1
Choose bₙ = 1/n² (dominant terms). ∑ 1/n² converges: p-series, p = 2
2
L = lim (2n³ + n²)/(n³ + n + 4) = 2
3
0 < 2 < ∞, so both series behave the same way
The series CONVERGES
Which Test Should I Use?
Integral Test
Limit Comparison
•
aₙ = f(n) with f easy to integrate
•
•
Needs: positive, continuous,
decreasing
aₙ looks like a p-series or geometric
series
•
Needs: aₙ > 0 and bₙ > 0
Examples: 1/(n ln n), n·e⁻ⁿ²
•
Examples: (2n+1)/(n³+n+4), 1/(2ⁿ−1)
•
Either way: check the conditions first. Neither test gives the sum.
Your Turn
Work individually. Show conditions or bₙ, computation, and conclusion. (15 min)
Integral Test
Limit Comparison
a) ∑ 1/(n ln n), n ≥ 2
a) ∑ (n²+3n)/(n⁴+1)
b) ∑ 1/(3n+1)²
b) ∑ (4n+1)/(n²+n+3)
c) ∑ n/eⁿ
c) ∑ sin(1/n)
Why can’t the Integral Test be used on ∑ (−1)ⁿ/n ?
Find the error: b = 1/n, L = 0 for ∑ 1/n², so “it
converges”.
Wrap-Up
Exit Ticket
Assignment
In 2–3 sentences: why must we verify
the conditions of a test before
concluding? Give one example from
today.
•
Integral Test: ∑ 1/(n²+4)
•
Limit Comparison: ∑ (5n²−n)/(n⁴+3)
•
Create your own series, choose a
test, solve it, and show a Desmos
graph or sketch