9.3 – Convergence of Series

Sigma summation notation:

𝑆 = 𝑎 =

Partial sums

𝑆 = 𝑎 =

𝑆 =

𝑆 =

𝑆 =

Limits and convergence vs divergence for series definition

If lim→

𝑆 = 𝑆, then ∑ 𝑎 = 𝑆 and converges.

If lim→

𝑆 does not exist, then the series diverges.

Graphing series – rectangles

Each rectangle’s area represents

Sum of area of first 𝑛 rectangles represents

Theorem 9.2: Convergence Properties of Series (4 Properties)

1. Sum Rule, Constant Multiple Rule for convergence

If  𝑎  and  𝑏  and if 𝑘 is a constant, then

𝑎.      (𝑎 + 𝑏 )  converges to  𝑎 + 𝑏

𝑏.      𝑘 ⋅ 𝑎  converges to 𝑘 ⋅ 𝑎

2. Changing a finite number of terms in a series does not change

whether or not it converges, although it may change the value

of the sum if it does converge (assuming no vertical

asymptotes)

3. 𝑛 Term Test:

If  lim→

𝑎 ≠ 0 or  𝑙𝑖𝑚→

𝑎  𝐷𝑁𝐸, then  𝑎  diverges

4. Constant Multiple Rule for divergence

If  𝑎  diverges, then  𝑘 ⋅ 𝑎  diverges if 𝑘 ≠ 0

Examples – Converge or diverge?

1.     3𝑛 + 2𝑛 − 45𝑛 − 7𝑛 + 1

2. Harmonic series (has to do with music)

1𝑛

3.     1𝑛

Theorem 9.3 – The Integral Test

Let 𝑎 = 𝑓(𝑛) where 𝑓(𝑥) is continuous, decreasing, and positive

If  𝑓(𝑥)𝑑𝑥  converges, then  𝑎  converges

If  𝑓(𝑥)𝑑𝑥  diverges, then  𝑎  diverges

𝑝-Series Test:

1𝑛

 converges if 𝑝 > 1 and diverges if 𝑝 ≤ 1

(Harmonic series is 𝑝-series with 𝑝 = 1)

Examples – Converge or diverge?

4.     4

(2𝑛 + 1)

5.     1

25 + 𝑛

6.     (ln 𝑛)

𝑛

7.     34

+512