Lecture Notes: Infinite Sums and Series

Introduction to Series

• Discussed infinite sums, known as series.
• Explored geometric series and telescoping series where answers can be found.
• The focus is on determining if a series converges (has a sum).

Convergence and Divergence

• Convergence of a Series: A series converges if the sum can be calculated.
• Divergence: If a series does not converge, it diverges.

Key Observations

• If a series converges, the individual terms must approach 0 as n approaches infinity.
• Contrapositive Statement: If the limit of terms is not 0, the series diverges.

Examples

1. Series of n (1+2+3+...): The nth term is n, which does not approach 0. Therefore, the series diverges.
2. Alternating Series (-1)^n: Terms oscillate between 1 and -1; hence, do not converge to 0. The series diverges.
3. Series with terms (-1/2)^n: Terms go to 0. However, convergence not guaranteed without further analysis.

Harmonic Series

• Definition: Sum of 1/n from n=1 to infinity.
• Using nth-term test: Limit of terms is 0, but does not imply convergence.
• Demonstrated that harmonic series diverges by grouping terms into blocks greater than 1/2.

Integral Test

• Purpose: Connects series and integrals to determine convergence/divergence.
• If f(x) is continuous, positive, and decreasing, then series and integral either both converge or both diverge.
• Error Estimation: Error in approximating sum using partial sums can be bounded by integral of the corresponding function.

Application of Integral Test

• Example: Show series of 1/(n^2 + 1) converges.
  • Integral test with f(x) = 1/(x^2 + 1) shows it converges (tied to arctangent function).

P-Series

• Definition: Series of form 1/n^p.
• Converges if p > 1, diverges if p ≤ 1.
• Important examples include the sum of 1/n^2 (converges to π²/6).

Calculating Specific Sums

• Problem: Find sum n * r^n for |r| < 1.
• Solution involves breaking series into smaller geometric series.
  • Result: Sum equals ( \frac{r}{(1-r)^2} ).*

Conclusion

• Series convergence is a complex topic that requires careful analysis using different tests and techniques.
• Tests like nth-term test and integral test provide frameworks for understanding the behavior of series.
• Understanding geometric series and p-series is crucial for solving more advanced problems in series analysis.