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Section 8.2: Infinite Series PowerPoint PPT Presentation


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Section 8.2: Infinite Series. Zeno’s Paradox. You can’t actually get anywhere because you always have to cover half the remaining distance! You have to do half, then half that.. etc. Can you add infinitely many numbers ??. Informal Definition A series is sequence added up.

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Section 8.2: Infinite Series

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Section 8 2 infinite series

Section 8.2: Infinite Series


Section 8 2 infinite series

Zeno’s Paradox

You can’t actually get anywhere because you always have to cover half the remaining distance!

You have to do half, then half that.. etc.

Can you add infinitely many numbers ??


Section 8 2 infinite series

Informal Definition

A series is sequence added up.


Section 8 2 infinite series

Formal Definition

A series converges

if the sequence of partial sums converges.


Section 8 2 infinite series

Example


Section 8 2 infinite series

Definition

is a geometric series.


Section 8 2 infinite series

Theorem

Diverges if

Convergesto if


Section 8 2 infinite series

Proof:

If and only if


Section 8 2 infinite series

This solves Zeno’s paradox!!


Section 8 2 infinite series

Diverges


Section 8 2 infinite series

Telescoping Series

Use Partial Fractions


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