Sigma notation Express the following sums using sigma notation. (Answers are not unique.)
(d) 1 + 1/2 + 1/3 + 1/4

Verified step by step guidance

1. Identify the pattern in the given sequence: The terms are fractions with the numerator fixed at 1 and the denominator increasing sequentially (1, 2, 3, 4).
2. Recognize that the general term for this sequence can be expressed as $\frac{1}{k}$, where $k$ represents the position of the term in the sequence.
3. Determine the range of the index $k$: The sequence starts at $k=1$ and ends at $k=4$.
4. Write the sum in sigma notation: $\sum k_1^4\frac{1}{k}$, where the summation symbol $\sum$ indicates the sum of the terms from $k=1$ to $k=4$.
5. Verify the sigma notation by expanding it: Substitute $k$ values (1, 2, 3, 4) into the general term $\frac{1}{k}$ to confirm that it matches the original sequence.

Key Concepts

Sigma Notation

Sigma notation is a concise way to represent the sum of a sequence of terms. It uses the Greek letter sigma (Σ) to indicate summation, followed by an expression that defines the terms to be added. The notation typically includes an index of summation, which specifies the starting and ending values for the variable that represents the terms.

Index of Summation

The index of summation is a variable used in sigma notation to denote the position of each term in the sequence being summed. It usually starts at a specified lower limit and increments by one until it reaches an upper limit. For example, in the sum Σ from i=1 to n, 'i' is the index that takes on values from 1 to n.

Harmonic Series

The series represented by the sum 1 + 1/2 + 1/3 + 1/4 is known as the harmonic series. Each term in this series is the reciprocal of a positive integer. Understanding the harmonic series is important because it illustrates concepts of convergence and divergence in infinite series, although in this case, we are only summing a finite number of terms.