Solutions: Mathematica 6 Lab 0

Problem 1:

?Binomial

Binomial[81,14]

Compare that with

((81!)/(14!(81-14)!))

(∑_(k=0)^n)Binomial[n,k]

Problem 2:

(∑_(k=0)^∞)(((-1))^k)(((2k+1))^(-2))

?Catalan

N[Catalan]
Catalan//N

N[Catalan,50]

Lab Number 0

Evaluated by Jeremy Silver

DateString[]

Problem 1

?Binomial

Binomial[81,14]

(∑_(k=0)^n)Binomial[n,k]

Problem 2

(∑_(k=0)^∞)(((-1))^k)(((2k+1))^(-2))

?Catalan

N[Catalan,50]

Hyperlink["Catalan Biography","http://en.wikipedia.org/wiki/Eug%C3%A8ne_Charles_Catalan"]

Problem 3

(∑_(k=0)^∞)(((-1))^k)(((2k+1))^(-1))

N[%,50]

Problem 4

Grid[{{Text[(π^π)],N[(π^π),20]},{Text[(E^π)],N[(E^π),20]},{Text[(π^E)],N[(π^E),20]},{Text[(E^E)],N[(E^E),20]}}]

(π^π)>(E^π)>(π^E)>(E^E)

Problem 5

(∑_(k=0)^∞)(x^k)

(∑_(k=i)^∞)(1/(((2k-1))^4))

(∑_(k=1)^∞)(1/((2^k)(k^2)))

(∏_(k=1)^∞)(1+((((-1))^(k+1))/(2k-1)))

x(∏_(k=1)^∞)(1-((x^2)/((k^2)(π^2))))

Use FullSimplify to get rid of the radicals. Automatic in v7.

Problem 6

∫(1/(1-(x^3)))\[DifferentialD]x
TraditionalForm[%]
TeXForm[%]

∫(1/(1-(x^5)))\[DifferentialD]x//TraditionalForm
∫(1/(1-(x^7)))\[DifferentialD]x//TraditionalForm

Problem 7

?Table

triangularnumbers=Table[(∑_(k=1)^n)k,{n,10}]

Table[(∑_(k=1)^n)(k^3),{n,10}]

(triangularnumbers^2)

(∑_(k=1)^n)(k^3)==((((∑_(k=1)^n)k))^2)

True

Hyperlink["CatalanBiography","http://en.wikipedia.org/wiki/Eugene_Charles_Catalan"]

Problem 3:

(∑_(k=0)^∞)(((-1))^k)(((2k+1))^(-1))

N[%,50]

Problem 4:

N[(π^π),20]

N[(E^π),20]

N[(π^E),20]

N[(E^E),20]

N[(π^π)]>N[(E^π)]>N[(π^E)]>N[(E^E)]

Problem 5:

(∑_(k=0)^∞)(x^k)

Note that Mathematica did not report the convergence condition.

(∑_(k=1)^∞)(1/(((2k-1))^4))

(∑_(k=1)^∞)(1/((2^k)(k^2)))

In the remaining exercises it is important to remember that Mathematica reserves curly brackets { } to identify lists:

(∏_(k=1)^∞)(1+((((-1))^(k+1))/(2k-1)))

x(∏_(k=1)^∞)(1-((x^2)/((k^2)(π^2))))

Simplify[%,x>0]

Problem 6:

∫(1/(1-(x^3)))\[DifferentialD]x

TraditionalForm[%]

∫(1/(1-(x^5)))\[DifferentialD]x

∫(1/(1-(x^7)))\[DifferentialD]x

∫(1/(1-(x^7)))\[DifferentialD]x//TraditionalForm

Problem 7:

?Table

triangularnumbers=Table[(∑_(k=1)^n)k,{n,1,10}]

sumofcubes=Table[(∑_(k=1)^n)(k^3),{n,1,10}]

(∑_(k=1)^n)(k^3)===((((∑_(k=1)^n)k))^2)