convergence of series sigma summation...
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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.
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Graphing series – rectangles
Each rectangle’s area represents
Sum of area of first 𝑛 rectangles represents
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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)
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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
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Examples – Converge or diverge?
1. 3𝑛 + 2𝑛 − 45𝑛 − 7𝑛 + 1
2. Harmonic series (has to do with music)
1𝑛
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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
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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)
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5. 1
25 + 𝑛
6. (ln 𝑛)
𝑛
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7. 34
+512