Series 2

What is the necessary condition for the convergence of a series Σaₙ?

If Σaₙ converges, then aₙ → 0 as n → ∞.

What is the contrapositive of the necessary condition for convergence?

If aₙ does not tend to 0, then the series Σaₙ does not converge.

Does aₙ → 0 guarantee that Σaₙ converges?

No. It is necessary but not sufficient: a series may still diverge even though its terms tend to 0.

Why must aₙ → 0 when Σaₙ converges?

If sₙ is the sequence of partial sums, then aₙ = sₙ₊₁ − sₙ. If sₙ → S, then aₙ → S − S = 0.

What does it mean for a series to have eventually non-negative terms?

There exists an index N such that aₙ ≥ 0 for every n ≥ N.

What is the regularity theorem for series with eventually non-negative terms?

Every series with eventually non-negative terms is either convergent or positively divergent to +∞.

When does a series with eventually non-negative terms converge?

It converges if and only if its sequence of partial sums is bounded above.

Why are the partial sums of a series with non-negative terms increasing?

Because sₙ₊₁ = sₙ + aₙ₊₁ and aₙ₊₁ ≥ 0, so sₙ₊₁ ≥ sₙ.

What happens if the partial sums of a non-negative-term series are unbounded above?

The partial sums tend to +∞, so the series is positively divergent.

Why does changing or removing finitely many terms not affect a series’ character?

It changes the partial sums by only a fixed finite amount, so it cannot change convergence into divergence or vice versa.

What is the generalized harmonic series?

Σ from n=1 to ∞ of 1/n^α, where α ∈ ℝ.

When does the generalized harmonic series Σ from n=1 to ∞ of 1/n^α converge?

It converges when α > 1.

When does the generalized harmonic series Σ from n=1 to ∞ of 1/n^α diverge?

It diverges when α ≤ 1.

What is the generalized logarithmic harmonic series?

Σ from n=2 to ∞ of 1/[n^α(ln n)^β], where α, β ∈ ℝ.

When does Σ from n=2 to ∞ of 1/[n^α(ln n)^β] converge?

It converges when α > 1 and β > 0, and also when α = 1 and β > 1.

When does Σ from n=2 to ∞ of 1/[n^α(ln n)^β] diverge?

It diverges when α < 1 for every β ∈ ℝ, and also when α = 1 and β ≤ 1.

What does the generalized logarithmic harmonic series become when β = 0?

It becomes the generalized harmonic series Σ from n=2 to ∞ of 1/n^α.

What are the assumptions of the direct comparison criterion?

Σaₙ and Σbₙ have positive terms, and aₙ ≤ bₙ eventually.

What does the direct comparison criterion say about divergence?

If 0 < aₙ ≤ bₙ eventually and Σaₙ diverges to +∞, then Σbₙ also diverges to +∞.

What does the direct comparison criterion say about convergence?

If 0 < aₙ ≤ bₙ eventually and Σbₙ converges, then Σaₙ also converges.

Which series should be the larger one when using direct comparison to prove convergence?

The known convergent benchmark series should be the larger series: 0 ≤ aₙ ≤ bₙ and Σbₙ converges.

Which series should be the smaller one when using direct comparison to prove divergence?

The known divergent benchmark series should be the smaller series: 0 ≤ aₙ ≤ bₙ and Σaₙ diverges.

What does aₙ ~ bₙ mean?

It means lim as n → ∞ of aₙ/bₙ = 1; the sequences have the same dominant asymptotic behavior.

What is the asymptotic comparison criterion?

If aₙ > 0, bₙ > 0, and aₙ ~ bₙ, then Σaₙ and Σbₙ have the same character: either both converge or both diverge.

What does bₙ = o(aₙ) mean?

It means lim as n → ∞ of bₙ/aₙ = 0; bₙ is negligible compared with aₙ.

What proposition connects little-o notation with convergence?

If bₙ = o(aₙ) and Σaₙ converges, then Σbₙ converges.

What is the value of the exponential series Σ from n=0 to ∞ of 1/n!?

Σ from n=0 to ∞ of 1/n! = e.

What is the power-series representation of eˣ?

For every x ∈ ℝ, eˣ = Σ from n=0 to ∞ of xⁿ/n!.

What limit also represents eˣ?

For every x ∈ ℝ, eˣ = lim as n → ∞ of (1 + x/n)ⁿ.

How can a series with eventually non-positive terms be studied?

Rewrite Σaₙ as −Σ(−aₙ). Since −aₙ is eventually non-negative, use the criteria for non-negative series.

What does it mean for a series Σaₙ to converge absolutely?

Σaₙ converges absolutely if the series of absolute values Σ|aₙ| converges.

What is the absolute convergence criterion?

If Σ|aₙ| converges, then Σaₙ converges.

Does ordinary convergence imply absolute convergence?

No. A series may converge without converging absolutely.

What does it mean for a series to be conditionally convergent?

Σaₙ is conditionally convergent if Σaₙ converges but Σ|aₙ| diverges.

What is the alternating harmonic series?

Σ from n=1 to ∞ of (−1)ⁿ⁺¹/n = 1 − 1/2 + 1/3 − 1/4 + ⋯.

What is the value of the alternating harmonic series?

Σ from n=1 to ∞ of (−1)ⁿ⁺¹/n = ln 2.

Why is the alternating harmonic series conditionally convergent?

The original series converges to ln 2, but its absolute-value series is Σ from n=1 to ∞ of 1/n, which diverges.

What should be checked first when determining the character of Σaₙ?

Compute lim aₙ. If aₙ does not tend to 0, the series diverges; if aₙ → 0, further analysis is required.

Which methods should be considered for a series with eventually non-negative terms?

A benchmark series, the direct comparison criterion, or the asymptotic comparison criterion.

What should be done when the signs of a series keep changing?

First study Σ|aₙ|. If it converges, then Σaₙ converges absolutely and therefore converges.

What can be concluded if Σ|aₙ| diverges?

Nothing by itself: Σaₙ may diverge or may converge conditionally.