Study Notes on Infinite Sequences and Series and Integral Test (Chapter 9)

Chapter 9: Infinite Sequences and Series

9.3 The Integral Test

• The primary question regarding an infinite series is whether it converges.

• For series with nonnegative terms, convergence is determined by the boundedness of the sequence of partial sums.

• Nondecreasing Partial Sums:

  • For a series $extstyle ext{sum} \, (a_n)<em>{n=1}^{ ext{∞}}$ where $a_n ext{≥} 0 ext{ for all } n$, the sequence of partial sums $s_n = a_1 + a_2 + … + a_n$ is nondecreasing because each term added is nonnegative: $s</em>{n+1} = s_n + a_{n+1}$

  • Nondecreasing implies $s_1 ext{ ≤ } s_2 ext{ ≤ } …$, which allows the application of the Monotonic Sequence Theorem (Theorem 6, Section 9.1).

Corollary of Theorem 6

• A series $extstyle ext{sum} \, (a_n)_{n=1}^{ ext{∞}}$ with nonnegative terms converges if and only if its partial sums are bounded from above.

Example 1: The Harmonic Series

• The harmonic series $extstyle ext{sum}_{n=1}^{ ext{∞}} rac{1}{n} = 1 + rac{1}{2} + rac{1}{3} + …$ diverges despite individual terms tending to zero, because the partial sums are unbounded.

• By grouping terms:

  • Grouping strategy:

  • 1st Group: $1$ = 1

  • 2nd Group: $rac{1}{2}$ = 0.5

  • 3rd Group: $rac{1}{3} + rac{1}{4}$ > 0.5

  • 4th Group: $rac{1}{5} + rac{1}{6} + rac{1}{7} + rac{1}{8}$ > 0.5

• Continuing this pattern yields that every group contributes at least 0.5, hence showing unboundedness.

The Integral Test

• Introduces the Integral Test through series converging associated with the harmonic series:

• Example 2: Convergence of $extstyle ext{sum}_{n=1}^{∞} rac{1}{n^2}$:

  • Comparing with the integral:
    $extstyle ext{sum}_{n=1}^{∞} rac{1}{n^2} ext{ corresponds to } \, rac{1}{1} ext{(integrate) ≈ finite value.}$

• The series converges by showing the integral converges.

Theorem 9 — The Integral Test

• Let $ext{a_n} = f(n)$ where $f(x)$ is a continuous, positive, decreasing function for integer values of $n$. Then:

  1. $extstyle ext{sum}<em>{n=N}^{∞} a_n$ converges if $extstyle ext{integral}</em>{N}^{∞} f(x) dx$ converges.

  2. $extstyle ext{sum}<em>{n=N}^{∞} a_n$ diverges if $extstyle ext{integral}</em>{N}^{∞} f(x) dx$ diverges.

Proof Overview

1. For the case $N=1$, consider rectangles drawn under the curve. This shows that the areas of rectangles exceed the integral areas.

2. If integrated areas converge, the rectangle sums converge and vice versa for divergence.

Examples of p-Series

• Result: A series $extstyle ext{sum}_{n=1}^{∞} rac{1}{n^p}$ converges if p > 1 and diverges if $p ext{ ≤ } 1$.

  • If p < 1, the integral diverges by limit tests. Therefore, evaluates with bounds on $p$.

• Example: For $p = 1$ ($1/n$ harmonic series), it diverges (previous rationale).

• For p < 1, $p = 0.5$ (diverges) and p > 1 ($1/n^2$ (converges)).

Another Integral Test Example — $ext{sum}_{n=1}^{∞} rac{1}{n^2 + 1}$

• Appropriate function $a_n$ that is positive, continuous yields:
  $L_{1}^{∞} rac{1}{n^2 + 1} dx$ converges.

Estimating the Error in Convergent Series

• For most series, we cannot determine the sum directly; thus, error estimation is key when approximating these sums.

• Various tests suggest bounding remainder $R_n = S - s_n$ ensures accuracy in measurement and operational assumptions.

Error Boundestimation using Integrals

• If converging, evaluate error by finding $R_n ext{ with areas (integrals)}$ related variables.

• The error approximation utilizes Riemann sums compared to continuous functions for validity and preciseness.

Conclusion

• The Integral Test provides a significant basis for determining convergence in series with potential analyses.

• The difficulty but necessity of controlling error using effective strategies portrays critical bridge equations of convergence and divergence,

• Deriving infinite tests gives rigorous measures upon operational analyses over approximations of functions.
  The material synthesizes crucial calculus with practical analysis rendered towards defining series and integral convergence assessments effectively.

Appendix

• Multiple examples showcase how to implement various tests indicating convergence around functional behavior effectively. Each cross-reference draws insights and error analysis techniques through theoretical conceptual practices, entirely tracking empirical data and function patterning.
  Equations cover consistent representations and yield continuous growth through derivative behavior showcase, validating estimates of function approximations comprehensively.

• Further exploration unveils integration into function-assessment formulations while yielding significant implications for understanding dominating characteristics through each specified example and theorem applied.
  Balancing polynomial-based understanding through integral tests fortifies convergence structures effectively across operational functions.