7/25/2019 Power Series Notes
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STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
7/25/2019 Power Series Notes
2/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
Ram Theorem:
= < > then
converges.Ram Theorem: = < > then isthe set of all sub sequential limit.
7/25/2019 Power Series Notes
3/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
7/25/2019 Power Series Notes
4/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
7/25/2019 Power Series Notes
5/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
7/25/2019 Power Series Notes
6/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
7/25/2019 Power Series Notes
7/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
7/25/2019 Power Series Notes
8/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
Power series
7/25/2019 Power Series Notes
9/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
lim sup|| = lim|| 0 = lim(2) 0 =
2 0 = 2
Hence, = 2
7/25/2019 Power Series Notes
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STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
roblem Find the radius of convergence of the power series whose nth term is given by
1. where, = 1, = 32. , = , 4 , Sol 1. = 3 = 3+
lim sup|| = lim|| 0 = lim 312 0
/ = lim3 0
= 3
Therefore, = 3.2. lim sup|| =
{lim lim 4 }
= 4 = Therefore, = 3.Remark: taking sup of sub sequential limits.
7/25/2019 Power Series Notes
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STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
7/25/2019 Power Series Notes
12/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
7/25/2019 Power Series Notes
13/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
7/25/2019 Power Series Notes
14/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
ApostalApostal
7/25/2019 Power Series Notes
15/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
7/25/2019 Power Series Notes
16/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
7/25/2019 Power Series Notes
17/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
HW SolHW Sol
7/25/2019 Power Series Notes
18/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
7/25/2019 Power Series Notes
19/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
7/25/2019 Power Series Notes
20/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
7/25/2019 Power Series Notes
21/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
7/25/2019 Power Series Notes
22/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
Apostal NotesApostal Notes
7/25/2019 Power Series Notes
23/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
7/25/2019 Power Series Notes
24/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
Rudin NotesRudin Notes
7/25/2019 Power Series Notes
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STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
TrenchTrench
7/25/2019 Power Series Notes
26/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
7/25/2019 Power Series Notes
27/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
7/25/2019 Power Series Notes
28/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
7/25/2019 Power Series Notes
29/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
7/25/2019 Power Series Notes
30/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
7/25/2019 Power Series Notes
31/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
7/25/2019 Power Series Notes
32/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
BartleBartle
7/25/2019 Power Series Notes
33/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK
LangLang
7/25/2019 Power Series Notes
34/34
STEP I: 1. Fast paragraph readings, 2. Slow paragraph readings, 3. All paragraph readings
STEPII: test paragraphs concepts
STEPIII: write all the concepts
STEP IV: TAKE REVISIONS ONE IN 24 HRS AND IN A WEEK