Real Analysis Final

Theorem (Monotonic Subsequences)

Let (a_n)n=1 to inf be any sequence. Then there exists monotonic subsequences whose limits are limsup(a_n) and liminf(a_n).

Defn. (Subsequential Limit)

Let (a_n)n=1 to inf be a sequence. A subsequential limit is any extended real number (can be infinity) that is the limit of a subsequence of (a_n)n=1 to inf. Denoted by S.

Thm (S)

Let (a_n)n=1 to inf be any sequence of real numbers, and S the corresponding set of sequential limits of (a_n)n=1 to inf. Then we have the following:

1) S isn’t the empty set

2) supS=limsup(a_n) and infS=liminf(a_n)

3) lim n-to-inf a_n exists iff S has exactly one element, name the limit of (a_n)n=1 to inf.

Defn. 2.4.3 (Convergence of a Series)

Let (a_n)n=1 to inf be a sequence. An infinite series is a formal expression of the sums of the terms. We define the corresponding sequence of partial sums (s_m)m=1 to inf by

s_m = a_1 + a_2 + … + a_m,

and say the series of sums converges to A if the sequence of partial sums converges to A.

Thm 2.4.6 (Cauchy Condensation Test)

Suppose (a_n) is decreasing and satisfies a_n >= 0 for all n e N. Then, the series converges iff the series

SUM) 2^n*a_2n = a_1 + 2a_2 + 4a_4 + 8a_8 + 16a_16 + …

converges.

Corollary 2.4.7

The series

SUM) 1/n^p

converges iff p > 1.

Thm 2.7.1 (Algebraic Limit Thm for Series)

If the infinite series a_k converges to A and the infinite series b_k converges to B then

1) multiplying the series by a constant c equals cA which is equivalent to pulling the constant c out

2) adding a_k and b_k equals A+B which is equivalent to summing both a_k and b_k seperately.

Thm. 2.7.2 (Cauchy Criterion for Series)

The infinite series of a_k converges iff, given epsilon > 0, there exists an N as an element of the natural numbers such that whenever n>m>=N it follows that

|a_m+1 + a_m+2 + … + a_n| < epsilon.

Said another way, the series a_n converges iff its sequence of partial sums in Cauchy.

Another way, for all epsilon > 0, the exists N as an element of the natural numbers such that for all m,n>=N, |s_n - s_m| < epsilon.

Thm 2.7.3

If the infinite series of a_k converges, then a_n as n approaches infinity converges to 0.

Thm 2.7.4 (Comparison Test)

Assume (a_k)k=1 to inf and (b_k)k=1 to inf are sequences satisfying 0 <= a_k <= b_k for all kinds of. Then

1) if sum(b_k) converges, sum(a_k) converges

2) if sum(a_k) diverges, then sum(b_k) diverges.

Defn 2.7.8 (Absolute/Conditional Convergence)

If SUM(|a_n|) converges, then we say the original series converges absolutely.

If the original series converges but SUM(|a_n|) does not converge, then we say the original series converges conditionally.

Thm 2.7.6 (Absolute Convergence Test)

If the series SUM(|a_n|) converges, then the original series converges as well.

Thm (Root Test)

Let (a_n)n=1 to inf be a sequence, and let A=limsup(|a_n|^1/n). Then the series SUM(a_n)

1) converges absolutely if A<1

2) diverges if A>1

3) test gives no information if A=1

Thm (Ratio Test)

Let (a_n)n=1 to inf be a sequence. Then the series SUM(a_n)

1) coverges absolutely if limsup|a_n+1/a_n|<1

2) diverges if liminf|a_n+1/a_n|>1

3) test gives no information if liminf <= 1 <= limsup

Thm 2.7.7 (Alternating Series Test)

Let (a_n) be a sequence satisfying

1) terms are decreasing

2) (a_n) converges to 0

Then, the alternating series SUM(-1)^n*a_n converges.

Thm (Abel’s Test)

If

1) the infinite series of a_n coverges and

2) (b_n) is a bounded monotone sequence

then the infinite series of a_n*b_n coverges.