Understanding Infinite Series

• A series is created by adding the terms of an infinite sequence.
• Expressed as: u1 + u2 + u3 + ... + un + ... or Σ∞n=1 un or Σ un.

Partial Sum

• Denoted as sn, important if ∑∞n=1 un exists
• s1 = u1
• s2 = u1 + u2
• s3 = u1 + u2 + u3
• s4 = u1 + u2 + u3 + u4
• sn = u1 + u2 + u3 + ... + un = Σni=1 ui, which will form a new sequence {sn}

Convergent Series

• If sn approaches a finite unique limit s as n approaches infinity.
• Expressed as: lim n→∞ sn = s (finite); This value 's' is known as the sum of the series.

Divergent Series

• A series is divergent if sn approaches +∞ or does not have a limit as n approaches infinity.
• Expressed as: lim n→∞ sn = +∞, or the limit does not exist.

Geometric Series

• The series is a + ar + ar² + ar³ + ... + arn + ... = Σ∞n=1 arn-1, with a ≠ 0.
• When r = 1, sn = a + a + a + ... + a = na → +∞ as n → ∞, meaning there's no limit and the series diverges.
• Where r ≠ 1, sn = a + ar + ar² + ar³ + ... + arn-1 and rsn = ar + ar² + ar³ + ... + arn-1 + arn.
• Subtracting these equations gives sn - rsn = a - arn, allowing sn = a(1 - rn) / (1 - r)
• For -1 < r < 1, rn → 0 as n → ∞, so lim n→∞ sn = a / (1 - r)
• If |r| < 1, the geometric series converges, and its sum equals a / (1 - r).
• If r ≤ -1 or r > 1, the limit lim n→∞ rn does not exist, and the geometric series diverges.

Summary of Geometric Series

• The geometric series a + ar + ar² + ar³ + ... + arn + ... = Σ∞n=1 arn-1, where a ≠ 0.
• Converges if |r| < 1 with Sum: Σ∞n=1 arn-1 = a / (1 - r)
• Diverges if |r| ≥ 1.

Telescoping Series

• Series Σ∞n=1 1 / (n(n+1))
• It is convergent if sn = u1 + u2 + u3 + ... + un
• Which can be expressed as (1 - ½) + (½ - ⅓) + (⅓ - ¼) + ... + (1/n - 1/(n+1))
• Solving it leads to a solution of = 1 - 1/(n+1)
• The limit would approach = lim n→∞ (1 - 1/(n+1)) 1 - 0 = 1
• Therefore, the series is convergent, and its sum is 1

Theorem for Series

• If a series Σ∞n=1 un is convergent, then lim n→∞ un = 0.

Test for Divergence

• If lim n→∞ un ≠ 0, then the series ∑ un is divergent.
• If ∑ un and ∑ vn are two convergent series, then ∑ cun, ∑(un + vn), and ∑(un - vn) are also convergent.
• ∑cun = c∑un
• Σ(un + vn) = ∑un + ∑vn
• Σ(un – vn) = ∑un - ∑vn
• Removing or adding a finite number of terms from a series does not affect its convergence.

Positive Term Series

• Testing Method: Integral Test

Integral Test

• Given a continuous, positive decreasing function f on [1, ∞) where un = f(n).

• The series Σ∞n=1 un converges or diverges depending on whether the improper integral ∫∞1 f(x)dx is finite or infinite.

• If ∫∞1 f(x)dx is convergent (finite), then Σ∞n=1 un is convergent.

• If ∫∞1 f(x)dx is divergent (infinite), then Σ∞n=1 un is divergent.

• For solving the value of p

  • If p < 0, then lim n→∞ 1/np = ∞, making the series diverge
  • If p = 0, then lim n→∞ 1/np = 1, also causing divergence.
  • If p > 0 and f(x) = 1/xp is continuous and decreasing on [1, ∞), use the integral test.

Comparison Test

• Comparing two series, assuming Σun and Σvn are positive
• If Σvn is convergent and un ≤ vn for all n, then Σun is also convergent.
  • If Σvn is divergent and un ≥ vn for all n, then Σun is also divergent. Example
• Let's say we have a series =1
• By limit comparison with a test series = 0
• If a limit exists between test series and known convergent or divergent series then both series' converge or diverge

De' Alembert's Ratio Test

• For series ∑∞n=1 un with positive terms and un ≠ 0 for all n, if lim n→∞ |(un+1) / un| = L:
  • If 0 ≤ L < 1, the series converges.
  • If L > 1, the series diverges. -If L = 1, the test is inconclusive.

Raabe's Test

• Used when the Ratio Test fails.
• For a series, given ∑∞n=1 un with positive terms and un ≠ 0 for all n, and the condition that lim n→∞ n((un / un+1) - 1) = L, then:
  • If L > 1, the series converges.
  • If L < 1, the series diverges.
  • If L = 1, the test is inconclusive.

Cauchy's Root Test

If for all positive terms of a series 2, un ≠ 0 and lim un L then For 0 < L < 1, series converges • For L > 1, series diverges • For L = 1, test fails.

Alternating Series

• Series where terms alternate in sign (e.g., positive, negative, positive, negative...)

Leibnitz's Test for Alternating Series

• Convergence can also be verified with series ∑(-1)^n-1 un = u1 - u2 + u3 - u4 +u5 - u6 + , un > 0 (must meet requirements)
• lim , = 0 as n approaches ∞ (ii) +1 ≤ un all n

Absolute and Conditional Convergence

• Series converge absolutely if the infinite limit is convergent, and conditionally if it is divergent The series cun where the infinite limit is equal to En=1, then it means the
• series is positive, then will be absolutely convergent.
• Series can be divergent if they reach a certain limit, the more the series tends to approach that value

Theorems

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