Series Knowledge-Based Study

Study notes

1. What is a Series?

   In algebra, a sequence is an ordered list of numbers, denoted as $a_1, a_2, a_3, \dots, a_n$. When we add the terms of a sequence together, the resulting expression is called a series.

   A finite series is written as:

   $$S_n = a_1 + a_2 + a_3 + \dots + a_n$$

   Sigma Notation

   To write sums compactly, mathematicians use the Greek capital letter sigma, $\Sigma$. The expression:

   $$\sum_{k=1}^{n} a_k$$

   is read as "the sum of $a_k$ from $k = 1$ to $n$". Here:

   • $k$ is the index of summation.
   • 1 is the lower limit.
   • $n$ is the upper limit.
   • $a_k$ is the general term or formula.

   Example: Expand and evaluate $\sum_{k=1}^{4} (2k + 1)$:

   $$\sum_{k=1}^{4} (2k + 1) = (2(1)+1) + (2(2)+1) + (2(3)+1) + (2(4)+1) = 3 + 5 + 7 + 9 = 24$$

   Understanding Partial Sums

   An infinite series continues indefinitely:

   $$\sum_{k=1}^{\infty} a_k = a_1 + a_2 + a_3 + \dots$$

   To analyze the behavior of an infinite sum, we examine its sequence of partial sums, $\{S_n\}$:

   • $S_1 = a_1$
   • $S_2 = a_1 + a_2$
   • $S_3 = a_1 + a_2 + a_3$
   • $S_n = \sum_{k=1}^{n} a_k$

   Convergence and Divergence

   If the limit of the sequence of partial sums exists and is a finite number $S$:

   $$\lim_{n \to \infty} S_n = S$$

   we say that the infinite series converges to the sum $S$. If the limit does not exist or approaches $\pm\infty$, the series diverges.

   Sum of an Arithmetic Sequence

   An arithmetic sequence has a constant difference $d$ between consecutive terms: $a_n = a_1 + (n - 1)d$. An arithmetic series is the sum of the terms of an arithmetic sequence.

   Derivation of the Sum Formula

   Writing the sum forward and backward:

   $$S_n = a_1 + (a_1 + d) + \dots + (a_n - d) + a_n$$

   $$S_n = a_n + (a_n - d) + \dots + (a_1 + d) + a_1$$

   Adding both equations term-by-term yields $2S_n = n(a_1 + a_n)$. Hence, the sum of the first $n$ terms is:

   $$S_n = \frac{n(a_1 + a_n)}{2}$$

   Substituting $a_n = a_1 + (n - 1)d$ gives the alternate form:

   $$S_n = \frac{n}{2}\left(2a_1 + (n - 1)d\right)$$

   Example: Find the sum of the first 20 terms of the arithmetic sequence $4, 7, 10, 13, \dots$:

   Here, $a_1 = 4$, $d = 3$, and $n = 20$.

   $$S_{20} = \frac{20}{2}\left(2(4) + (20 - 1)(3)\right) = 10(8 + 57) = 10(65) = 650$$

   Finite Geometric Series

   In a geometric sequence, each term is multiplied by a common ratio $r$: $a_n = a_1 r^{n-1}$. The sum of the first $n$ terms is given by:

   $$S_n = \frac{a_1(1 - r^n)}{1 - r} \quad (r \neq 1)$$

   If $r = 1$, all terms are identical, and $S_n = n a_1$.

   Infinite Geometric Series

   Consider an infinite geometric series $\sum_{k=1}^{\infty} a_1 r^{k-1}$. If $|r| < 1$, then $\lim_{n \to \infty} r^n = 0$. Taking the limit of $S_n$:

   $$S = \lim_{n \to \infty} \frac{a_1(1 - r^n)}{1 - r} = \frac{a_1}{1 - r}$$

   • If $|r| < 1$, the infinite geometric series converges to $S = \frac{a_1}{1 - r}$.
   • If $|r| \geq 1$ and $a_1 \neq 0$, the series diverges and has no finite sum.

   Example: Evaluate $\sum_{k=1}^{\infty} 12\left(\frac{1}{3}\right)^{k-1}$:

   Here, $a_1 = 12$ and $r = \frac{1}{3}$. Since $|\frac{1}{3}| < 1$, the series converges:

   $$S = \frac{12}{1 - \frac{1}{3}} = \frac{12}{\frac{2}{3}} = 12 \cdot \frac{3}{2} = 18$$

   Linearity Properties of Finite Sums

   Sigma notation satisfies two fundamental linear properties for constants $c$:

   • Constant Multiple Rule: $\sum_{k=1}^{n} c \cdot a_k = c \sum_{k=1}^{n} a_k$
   • Sum/Difference Rule: $\sum_{k=1}^{n} (a_k \pm b_k) = \sum_{k=1}^{n} a_k \pm \sum_{k=1}^{n} b_k$
   • Constant Sum Rule: $\sum_{k=1}^{n} c = n \cdot c$

   Common Pitfalls to Avoid

   • Confusing the $n$-th term with the sum: Remember that $a_n$ represents the specific value of term $n$, whereas $S_n$ represents the accumulated total of the first $n$ terms.
   • Misidentifying the number of terms: When the lower limit is $k = m$ and the upper limit is $n$, the number of terms is $n - m + 1$, not $n - m$.
   • Applying the infinite sum formula when $|r| \geq 1$: The formula $S = \frac{a_1}{1 - r}$ is only valid when the common ratio satisfies $-1 < r < 1$.