Example Solutions for

Step-by-Step Proofs

Mathematical Induction

Prove a sum or product identity using induction step by step:

prove by induction sum of j from 1 to n = n(n+1)/2 for n>0
prove sum(2^i, {i, 0, n}) = 2^(n+1) - 1 for n > 0 with induction
prove by induction product of 1 - 1/k^2 with k from 2 to n = (n + 1)/(2 n) for n>1

Prove divisibility by induction:

using induction, prove 9^n-1 is divisible by 4 assuming n>0
induction 3 divides n^3 - 7 n + 3

Derive a proof by induction of various inequalities step by step:

show with induction 2n + 7 < (n + 7)^2 where n >= 1
prove by induction (3n)! > 3^n (n!)^3 for n>0

Prove a sum identity involving the binomial coefficient using induction:

prove by induction sum C(n,k) x^k y^(n-k),k=0..n=(x+y)^n for n>=1
prove by induction sum C(n,k), k=0..n = 2^n for n>=1

Approximate the bounds of harmonic numbers using induction:

prove by induction sum 1/k from k=1 to 2^n >= 1 + n/2 for n>=0
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Trigonometric Identities

See the steps toward proving a trigonometric identity:

does sin(θ)^2 + cos(θ)^2 = 1?

Prove equalities between tangent, sine and cosine:

(1 + tan(x))/(1 - tan(x)) = (cos(x) + sin(x))/(cos(x) - sin(x))
cot(t/2)^2 = (1 + cos(t)) / (1 - cos(t))

Show the proof for equalities between trigonometric functions like tangent, arctangent, cosecant and secant:

verify tanθ + cotθ = secθ cscθ
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Series Convergence

Show how to prove if a sum of infinite terms diverges or converges with different tests: limit test, ratio test, root test, integral test, p-series test or geometric series test.

Investigate the convergence or divergence of an infinite sum step by step:

sum convergence of n
sum convergence of n^(-2)
does sum n/(n+2) converge
Does the sum of 2^(-n) converge?
Does the sum of 5*3^(1 - n) converge?
does sum 1/2^n - 1/2^(n+1) converge
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