Series notation simplifies complex sums using symbols like sigma (Σ). It's a shorthand for both finite and infinite series, with rules for manipulating terms. Understanding this notation is key to working with sequences and series efficiently.

Arithmetic and geometric series have specific formulas for calculating sums. These formulas are crucial in many real-world applications, from finance to physics. Knowing when series converge or diverge helps in solving problems involving infinite sums.

Series Notation and Formulas

Summation notation for series

• Summation notation uses the Greek letter sigma ($\sum$) to represent the sum of a series of terms concisely
  • Lower limit of summation written below $\sum$ symbol, upper limit written above
  • Index of summation is the variable used in summation (usually $i$ or $n$)
  • General term of series depends on index of summation, written to right of $\sum$ symbol
• Represents both finite series with fixed number of terms and infinite series that continue indefinitely
• Properties include constant multiple rule ($\sum_{i=1}^{n} ca_i = c\sum_{i=1}^{n} a_i$), sum rule ($\sum_{i=1}^{n} (a_i + b_i) = \sum_{i=1}^{n} a_i + \sum_{i=1}^{n} b_i$), and difference rule ($\sum_{i=1}^{n} (a_i - b_i) = \sum_{i=1}^{n} a_i - \sum_{i=1}^{n} b_i$)
• The sum of the first n terms of a series is called a partial sum

Arithmetic series sum formula

• Arithmetic series has constant difference between consecutive terms called common difference ($d$)
• General term of arithmetic series: $a_n = a_1 + (n-1)d$, where $a_1$ is first term and $n$ is term number
• Sum of first $n$ terms calculated using formula: $S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}[2a_1 + (n-1)d]$, where $S_n$ is sum of first $n$ terms
• Alternative formula for sum of arithmetic series: $S_n = n[a_1 + \frac{(n-1)d}{2}]$

Finite geometric series sums

• Geometric series has constant ratio between consecutive terms called common ratio ($r$)
• General term of geometric series: $a_n = a_1r^{n-1}$, where $a_1$ is first term and $n$ is term number
• Sum of first 1. $n$ terms of finite geometric series calculated using formula: $S_n = \frac{a_1(1-r^n)}{1-r}$, where $S_n$ is sum of first $n$ terms and $r \neq 1$
• If $|r| < 1$, geometric series converges to finite sum as $n$ approaches infinity: $S_\infty = \frac{a_1}{1-r}$, where $S_\infty$ is sum of infinite geometric series

Convergent infinite geometric series

• Infinite geometric series converges if and only if $|r| < 1$, where $r$ is common ratio; diverges if $|r| \geq 1$
• Sum of convergent infinite geometric series: $S_\infty = \frac{a_1}{1-r}$, where $a_1$ is first term and $r$ is common ratio
• Steps to determine convergence:
  1. Identify common ratio $r$
  2. Check if $|r| < 1$; series converges if true, diverges if false
  3. If series converges, calculate sum using formula $S_\infty = \frac{a_1}{1-r}$

Annuities and compound interest applications

• Annuity is series of equal payments made at regular intervals
  • Present value of annuity is sum of present values of all future payments
  • Future value of annuity is sum of all payments plus accumulated compound interest
• Compound interest formula calculates future value of lump sum investment: $A = P(1+\frac{r}{n})^{nt}$, where $A$ is future value, $P$ is principal, $r$ is annual interest rate, $n$ is number of compounding periods per year, and $t$ is number of years
• Present value of annuity formula: $PV = PMT \cdot \frac{1 - (1+\frac{r}{n})^{-nt}}{\frac{r}{n}}$, where $PV$ is present value, $PMT$ is periodic payment, $r$ is annual interest rate, $n$ is number of compounding periods per year, and $t$ is number of years
• Future value of annuity formula: $FV = PMT \cdot \frac{(1+\frac{r}{n})^{nt} - 1}{\frac{r}{n}}$, where $FV$ is future value, $PMT$ is periodic payment, $r$ is annual interest rate, $n$ is number of compounding periods per year, and $t$ is number of years

Special Series Types

• Telescoping series: A series where terms cancel out, leaving only a finite number of terms
• Harmonic series: A divergent series of the form $\sum_{n=1}^{\infty} \frac{1}{n}$
• Power series: An infinite series of the form $\sum_{n=0}^{\infty} a_n(x-c)^n$, where $a_n$ are coefficients and $c$ is a constant
• These special series types have unique properties and applications in various mathematical contexts