Lecture on Alternating Series Remainder

Concept of Error Approximations

• Error approximations involve estimating the sum of a series using the first n terms.
• The error is the leftover part when only a partial sum is used.
• In Taylor polynomials, using only a portion of the series results in an estimate that has some error (remainder).

Alternating Series Remainder

• The error in an alternating series is less than or equal to the absolute value of the next term after the terms used in the estimation.
• Example: Estimating with the first 6 terms involves calculating the partial sum and determining the potential error by examining the next term.

Example 1: Estimating a Series

• Approximate the sum using the first 6 terms:
  • 1 + (-1/2) + (1/3!) - (1/4!) + (1/5!) - (1/6!)
• Calculation results in approximately 0.6319.
• Next term calculation (1/7!) gives error bound: 0.000198.

Finding Upper and Lower Bounds

• The actual sum is between the calculated approximation and the approximation plus the error bound.
• Lower bound: Calculated sum (0.6319)
• Upper bound: Calculated sum + error (0.6319 + 0.000198)

Example 2: Error Less Than 0.001

• Objective is to find terms where error is less than 0.001.
• Continue adding terms until error term (next unused term) is less than 0.001.
• Example yields that using up to 1/8! provides adequate error control.

Applying Elementary Series

• Example with sigma notation and cosine of one:
  • Series: (-1)^n / (2^n * (2n)!)
  • Calculator approximation: 0.54030*

General Approach

• For alternating series, error is less than or equal to the absolute value of the first unused term.
• Formula for error in alternating series:
  • Error ≤ f(n+1)(x - c)^(n+1) / (n+1)!

Non-Alternating Series

• If the series doesn't alternate, use a different method:
  • Lagrangian error or Taylor's theorem remainder (to be covered in another session).

Conclusion

• Alternating series provide an accessible method for estimating series with controlled error.
• For non-alternating series, further methods like Lagrangian error are necessary.

Note: Further understanding requires familiarity with terms like factorial (!), sigma notation (Σ), and Taylor series.