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SERIES

A series is the sum of the elements of a sequence.

{an}n∈ℕ, {Sn}n∈ℕ is a sequence of nth partial sums , Sn = nΣk=0 ak, {S0 = a0, S1 = a0+a1, S2 = a0+a1+a2, ..., Sn = a0+a1+...+an = Sn-1+an}

{an}n∈ℕ, nΣk=0 ak: 1) it converges if ∃ limn→∞ Sn = S ∈ ℝ, Σk=0 ak = S that is the sum of the series; 2) it diverges if limn→∞ Sn = ±∞; 3) it is irregular if ∄ limn→∞ Sn, that is {Sn}n∈ℕ is irregular

+∞Σk=0 ak = limn→∞ nΣk=0 ak = S

A geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms.

Geometric series: +∞Σk=0 qk, q ∈ ℝ

+∞Σk=0 qk = +∞Σn=0 qn

Sn = nΣk=0 qk = (1-qn+1)/(1-q), q ≠ 1

Sn = nΣk=0 1k = n+1, q = 1

limn→∞ qn = {+∞, q > 1; 1, q = 1; 0, |q| < 1; ∄, q ≤ -1}; q ≠ 1, limn→∞ Sn = limn→∞ (1-qn+1)/(1-q) = {+∞, q > 1; 1/(1-q), |q| < 1; ∄, q ≤ -1}; q = 1, limn→∞ Sn = limn→∞ n+1 = +∞

q = 1/2, +∞Σk=0 (1/2)k = 1+1/2+1/4+1/8+1/16+...; +∞Σk=0 (1/2)k = 1/(1-1/2) = 2

q = 3, +∞Σk=0 3k = 1+3+9+27+81 = +∞

+∞Σk=0 (-1)k = 1-1+1-1+...; S0 = 0Σk=0 (-1)k = 1; S1 = 1Σk=0 (-1)k = 1-1 = 0; S2 = 2Σk=0 (-1)k = 1-1+1 = 1

Generalized harmonic series: +∞Σk=1 1/kα = +∞Σk=1 k, α ∈ ℝ

Harmonic series: +∞Σk=1 1/k = +∞

Sn = nΣk=1 1/k = 1+1/2+1/3+1/4+...+1/n

+∞Σk=1 1/k = 1+(1/2)+(1/3+1/4)+(1/5+1/6+1/7+1/8)+(1/9+1/10+1/11+1/12+1/13+1/14+1/15+1/16)+... ≥ 1+1/2+1/2+1/2+1/2+...; 1/3+1/4 ≥ 1/4+1/4 = 2/4 = 1/2, 1/5+1/6+1/7+1/8 ≥ 1/8+1/8+1/8+1/8 = 4/8 = 1/2, ...; it can be proved by induction that S2n ≥ 1+n/2 → +∞

A telescoping series is a series whose general term tn can be written as tn = an-an+1

The Mengoli series is a telescopic series: +∞Σk=1 1/k(k+1) = 1/1⋅2+1/2⋅3+1/3⋅4+...; Sn = nΣk=1 1/k(k+1) = nΣk=1 (1/k-1/(k+1)) = (1-1/2)+(1/2-1/3)+(1/3-1/4)+...+(1/(n-1)-1/n)+(1/n-1/(n+1)) = 1-1/(n+1) ⇒ Sn = 1-1/(n+1); limn→∞ Sn = 1 ⇒ +∞Σk=1 1/k(k+1) = 1

+∞Σk=0 ak, +∞Σk=n ak, these two series have the same behavior even if the sum could be different.

+∞Σk=1 qk, |q| < 1; +∞Σk=0 qk = 1/(1-q); +∞Σk=1 qk = +∞Σk=0 qk-q0 = 1/(1-q)-1 = 1/(1-q)-(1-q)/(1-q) = q/(1-q)

It is important to study methods to understand the behavior of the series; behavior means convergence, divergence or irregularity.

Necessary but not sufficient condition for convergence: if the series +∞Σk=0 ak is convergent, then limn→∞ an = 0

By hypothesis: limn→∞ Sn = S ∈ ℝ ⇒ limn→∞ Sn-1 = S; limn→∞ (Sn-Sn-1) = limn→∞ Sn - limn→∞ Sn-1 = 0; Sn-Sn-1 = (a0+a1+...+an-1+an)-(a0+a1+...+an-1) = an; limn→∞ an = limn→∞ (Sn-Sn-1) = 0

+∞Σk=0 ak = S ∈ ℝ ⇒ limn→∞ an = 0; limn→∞ an = 0 !⇒ +∞Σk=0 ak = S ∈ ℝ; limn→∞ 1/n = 0, +∞Σk=1 1/k = +∞

A necessary condition is useful only when it is not verified; if it is verified I cannot say anything about the corresponding property.

+∞Σk=1 (1+1/k)2; an = (1+1/n)2; limn→∞ an = 1 ⇒ +∞Σk=1 (1+1/k)2 does not converge.

+∞Σk=0 sin(k); ∄ limn→∞ sin(n) ⇒ +∞Σk=0 sin(k) does not converge; if the necessary condition is not verified, the series does not converge.

+∞Σk=0 sin(1/k); limn→∞ sin(1/n) = 0; if the necessary condition is verified, it is not possible to predict the behavior of the series, that is, the series could converge, diverge or be oscillating.

Positive term series: +∞Σk=0 ak, an ≥ 0 ∀ n ∈ ℕ

If +∞Σk=0 ak is a positive term series, then {Sn}n∈ℕ is increasing and therefore regular.

Sn+1 = a0+...+an+an+1 = Sn+an+1 ≥ Sn ⇒ Sn+1 ≥ Sn ∀ n ∈ ℕ

A positive term series is not irregular, so Sn is an increasing sequence therefore converges or diverges to +∞; an increasing monotone sequence converges to its supremum which can be a real number or + ∞.

+∞Σk=1 1/k2 is a positive term series.

+∞Σk=1 (-1)k/k is not a positive term series.

A positive term series is always regular, it is convergent or divergent to +∞.

There are 4 criteria to understand the behavior of a positive term series: comparison, asymptotic comparison, ratio, root.

Comparison criterion: +∞Σk=0 ak, +∞Σk=0 bk are two positive term series | ak ≤ bk ∀ k ∈ ℕ. Then, 1) +∞Σk=0 ak = +∞ ⇒ +∞Σk=0 bk = +∞, 2) +∞Σk=0 bk < +∞ ⇒ +∞Σk=0 ak < +∞

= +∞ means that it diverges; < +∞ means that it converges

ak ≤ bk ∀ k ∈ ℕ ⇒ nΣk=0 aknΣk=0 bk; if limn→∞ nΣk=0 ak = +∞ ⇒ limn→∞ nΣk=0 bk = +∞

If +∞Σk=0 ak < +∞, it is impossible to predict the behavior of +∞Σk=0 bk; if the minorant series converges, it is impossible to predict the behavior of the majorant series.

If +∞Σk=0 bk = +∞, it is impossible to predict the behavior of +∞Σk=0 ak; if the majorant series diverges, it is impossible to predict the behavior of the minorant series.

If the minorant series diverges, or if the majorant series converges, it is possible to understand the behavior of the series.

If the minorant series converges, or if the majorant series diverges, it is not possible to understand the behavior of the series.

+∞Σk=0 1/k! = 1/0!+1/1!+1/2!+1/3!+1/4!+...; if the necessary condition is not verified, the series does not converge and therefore diverges to +∞; the necessary condition is verified because limn→∞ 1/n! = 1/∞ = 0, so we need more information to understand the behavior of the series; k! = 1⋅2⋅3⋅...⋅k ≥ 1⋅2⋅2⋅...⋅2 ⇒ k! ≥ 2k-1 ⇔ 1/k! ≤ 1/2k-1; +∞Σk=0 1/2k-1 = +∞Σk=0 2/2k = 2 +∞Σk=0 (1/2)k, it is a geometric series with q = 1/2; +∞Σk=0 qk = {+∞, q ≥ 1; 1/(1-q), |q| < 1; ∄, q ≤ -1}; the majorant series is convergent and therefore, for the comparison criterion, the series converges, +∞Σk=0 1/k! < +∞; +∞Σk=0 1/k! = e

Asymptotic comparison criterion: +∞Σk=0 ak, +∞Σk=0 bk are two positive term series; 1) if limk→∞ ak/bk = l ≠ 0, then +∞Σk=0 ak < +∞ ⇔ +∞Σk=0 bk < +∞, the two series have the same behavior; 2) if limk→∞ ak/bk = 0, then +∞Σk=0 bk < +∞ ⇒ +∞Σk=0 ak < +∞, +∞Σk=0 ak = +∞ ⇒ +∞Σk=0 bk = +∞

+∞Σk=1 1/kα, α ∈ ℝ, generalized harmonic series; α = 1, +∞Σk=1 1/k = +∞; α = 2, +∞Σk=1 1/k2; 1/k2 ~ 1/k(k+1) ⇔ limk→∞ (1/k2)/(1/k(k+1)) = 1; limk→∞ (1/k2)/(1/k(k+1)) = limk→∞ k(k+1)/k2 = limk→∞ k2/k2 = 1; +∞Σk=1 1/k(k+1) = 1, telescopic series; considering the asymptotic comparison criterion with ak = 1/k2 and bk = 1/k(k+1), then +∞Σk=1 1/k2 < +∞

α ≥ 2, kα ≥ k2 ⇔ 1/kα ≤ 1/k2+∞Σk=1 1/kα+∞Σk=1 1/k2 < +∞; considering the criterion of comparison +∞Σk=1 1/kα < +∞ ∀ α ≥ 2

α ≤ 1, kα ≤ k ⇒ 1/k ≤ 1/kα ⇒ 1/kα ≥ 1/k ⇒ +∞Σk=1 1/kα+∞Σk=1 1/k = +∞; considering the criterion of comparison +∞Σk=1 1/kα = +∞

+∞Σk=1 1/kα {< +∞, α > 1; = +∞, α ≤ 1}

+∞Σk=1 1/k3/2 < +∞, α = 3/2 > 1

+∞Σk=1 1/k2/5 = +∞, α = 2/5 < 1

+∞Σk=1 sin(1/k), limk→∞ sin(1/k) = 0, the necessary condition is verified; sin(1/k) ≥ 0 ∀ k, it is a positive term series so it converges or diverges; sin(an) ~ an if limn→∞ an = 0, sin(an) ~ an ⇔ limn→∞ sin(an)/an = 1; sin(1/k) ~ 1/k for k → ∞, sin(1/k) ⇔ limk→∞ sin(1/k)/(1/k) = 1; +∞Σk=1 1/k = +∞ ⇒ for the asymptotic comparison, +∞Σk=1 sin(1/k) = +∞

Ratio criterion: +∞Σk=0 ak is a positive term series that is ak ≥ 0; if ∃ limk→∞ ak+1/ak = l ∈ ℝ ∪ {+∞}, then: 1) if l < 1 ⇒ +∞Σk=0 ak < +∞, 2) if l > 1 ⇒ +∞Σk=0 ak = +∞, 3) if l = 1 continue studying

Root criterion: +∞Σk=0 ak is a positive term series that is ak ≥ 0; if ∃ limk→∞ kak = l ∈ ℝ ∪ {+∞}, then: 1) if l < 1 ⇒ +∞Σk=0 ak < +∞, 2) if l > 1 ⇒ +∞Σk=0 ak = +∞, 3) if l = 1 continue studying

Considering the Cesàro criterion, if ∃ limk→∞ ak+1/ak = l ⇒ limk→∞ kak = l

+∞Σk=0 xk/k!, x ∈ ℝ, x > 0; it is a positive term series, limk→∞ xk/k! = 0, the necessary condition is verified; ak = xk/k!, limk→∞ ak+1/ak = limk→∞ (xk+1/(k+1)!)/(xk/k!) = limk→∞ (xk+1/(k+1)!)(k!/xk) = limk→∞ (xk⋅x/k!(k+1))(k!/xk) = limk→∞ x/(k+1) = 0 = l ⇒ for the ratio criterion, +∞Σk=0 xk/k! < +∞, ∀ x ∈ ℝ, x > 0

+∞Σk=1 xk/kk, x ∈ ℝ, x > 0; it is a positive term series, limk→∞ xk/kk = 0, the necessary condition is verified; ak = xk/kk, limk→∞ kak = limk→∞ kxk/kk = limk→∞ x/k = 0 ⇒ for the root criterion,+∞Σk=1 xk/kk < +∞

For the ratio criterion and the root criterion, if l = 1 continue studying; in this example the ratio criterion if l = 1; +∞Σk=1 1/k = +∞, ak = 1/k, limk→∞ ak+1/ak = limk→∞ (1/(k+1))/(1/k) = limk→∞ k/(k+1) = 1; +∞Σk=1 1/k2 < +∞, ak = 1/k2, limk→∞ ak+1/ak = limk→∞ (1/(k+1)2)/(1/k2) = limk→∞ k2/(k+1)2 = 1

Alternating series: +∞Σk=0 (-1)k·ak, ak ≥ 0 ∀ k ∈ ℕ

+∞Σk=0 (-1)k·ak = a0-a1+a2-a3+...

+∞Σk=1 1/k = 1+1/2+1/3+1/4+...

+∞Σk=1 (-1)k·1/k = -1+1/2-1/3+1/4+...

Leibniz criterion: +∞Σk=0 (-1)k⋅ak if, 1) limk→∞ ak = 0, 2) ak+1 < ak ∀ k ∈ ℕ, then +∞Σk=0 (-1)kak < +∞

+∞Σk=1 (-1)k1/kα α ∈ ℝ, generalized harmonic series with alternating signs; ak = 1/kα; 1) limk→∞ 1/kα = 0 if α > 0; 2) ak+1 < ak ⇔ 1/(k+1)α < 1/kα ⇔ kα < (k+1)α, true ∀ k

+∞Σk=1 (-1)k1/kα converges if α > 0

+∞Σk=1 1/kα converges if α > 1

Positive term series: comparison criterion, asymptotic comparison criterion, ratio criterion, root criterion.

Alternating series: Leibniz criterion.

+∞Σk=0 ak absolutely converges if +∞Σk=0 |ak| converges.

If +∞Σk=0 ak absolutely converges, then it converges.

|a+b| ≤ |a|+|b|; |nΣk=0 ak| ≤ nΣk=0 |ak|; |Sn| ≤ nΣk=0 |ak|

Absolute convergence is a sufficient condition but not a necessary condition for the convergence of the series.

If the series converges absolutely, then it converges, but it can also converge without converging absolutely.

Absolute convergence implies convergence, viceversa a series can converge without converging absolutely.

Absolute convergence ⇒ convergence

Convergence !⇒ absolute convergence

Convergence does not imply absolute convergence; a series can converge without converging absolutely.

The alternating harmonic series +∞Σk=1 (-1)k1/k converges for the Leibniz criterion, but it does not converge absolutely; +∞Σk=1 |(-1)k1/k| = +∞Σk=1 |(-1)k||1/k| = +∞Σk=1 1/k = +∞

+∞Σk=0 |ak| is a positive term series; if limk→∞ kak = l < 1, then +∞Σk=0 |ak| converges ⇒ +∞Σk=0 ak converges absolutely, then converges.