Discrrete mathematics for computer science series

Geometric SeriesBinomial Coefficients Harmonic Series... Sum of a Geometric Series• What is • Method 1: Prove by induction that for every n≥0, • And then make some argument about the lim

Geometric SeriesBinomial Coefficients

Harmonic Series

Sum of a Geometric Series

• What is

• Method 1: Prove by induction that for every n≥0,

• And then make some argument

about the limit as n →∞ to conclude that the sum is 2

Sum of a Geometric Series

• Another way Recall that

Manipulating Power Series

Identities involving “Choose”

• What is

• “Set Theory” derivation

– Let S be a set of size n

– This is the sum of the number of 0

element subsets, plus the number of element subsets, plus …, plus the

1-number of n-element subsets

– Total 2 n

n i

Using the Binomial Theorem

Using the Binomial Theorem

Stacking books

Can you stack identical books so the top one is completely off the table?

3/22/19

Stacking books

• Two books: balance top one at the

middle, the second over the table edge

at the center of gravity of the pair

Stacking books

the second over the third at ½, and the trio over the table edge at the center of gravity

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1

½

Stacking books

 Three books: balance top one at the

middle, the second over the third at ½, and the trio over the table edge at the center of gravity = (1+2+2½)/3 = 1⅚ from right end

3/22/19

1

½

1⅚

Stacking books

second over the third at ½, and the trio over the table edge at the center of gravity = (1+2+2½)/3

= 1⅚ from right end

The Harmonic Series Diverges

• Let Then for any n,

• Doubling the number of terms adds

at least ½ to the sum

• Corollary The series diverges, that is,

Proof by WOP that the Harmonic

The Harmonic Series Diverges

So with a stack of 31 books you get 2 book lengths off the table!

And there is no limit to how far the

stack can extend!

3/22/19

http://mathforum.org/advanced/robertd/harmonic.html

FINIS