Exam Questions - Series

Exam Questions - Series

Understanding Series

• A series in mathematics is the summation of a sequence of numbers or operations.
• Arithmetic series has a common difference between terms, while a geometric series has a common ratio.
• The sigma notation (Σ) is commonly used to represent a series, where a variable (like n) ranges over some set of values.

Series in Algebraic Form

• An arithmetic series can be represented algebraically as a + (a+d) + (a+2d) + … + (a+nd) where a is the first term and d is the common difference.
• A geometric series can be represented as a + ar + ar² + … + arⁿ where a is the first term and r is the common ratio.

Summation of Series

• The sum of an arithmetic series can be calculated using the formula S_n = n/2(2a + (n-1)d), where S_n is the sum of the first n terms.

• The sum to n terms of a geometric series can be evaluated using S_n = a (1 - r^n) / (1 - r) when

  r

  < 1, or S_n = a (r^n - 1) / (r - 1) when r > 1.

• It’s important to recognise which series type you’re dealing with to apply the correct sum formula.

Convergence of Series

• A series can either be convergent, where it approaches a finite number, or divergent, where it approaches infinity or has no limit.

• For geometric series, if the common ratio **

  r

  < 1**, the series is convergent, but if the **

  r

  >= 1**, the series is divergent.

• Convergence is a key concept when working with infinite series.

Key Strategies for Series Questions

• Always double-check whether the series is arithmetic or geometric by looking for a common difference or ratio.
• Be careful when dealing with negative values or fractions - small arithmetic errors can have big impacts on the final sum.
• Understand the convergence of a series. If the series is divergent, the sum to infinity is undefined.
• Practice questions with varying series types and term numbers to build your fluency in calculation.