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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

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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

Ram Theorem:

= < > then

converges.Ram Theorem: = < > then isthe set of all sub sequential limit.

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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

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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

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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

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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

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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

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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

Power series

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

lim sup|| = lim|| 0 = lim(2) 0 =

2 0 = 2

Hence, = 2

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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.

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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

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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

ApostalApostal

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

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

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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

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

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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

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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

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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

Apostal NotesApostal 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

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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

Rudin NotesRudin 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

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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

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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

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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

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

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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

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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

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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

BartleBartle

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

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