13.4 Series and Their Notations

Series notation simplifies complex calculations, making it easier to work with large sums. It's a powerful tool for representing arithmetic and geometric sequences, allowing us to compute sums efficiently using formulas tailored to each type.

Series have real-world applications in finance, population growth, and resource management. Understanding how to calculate sums and apply series formulas to practical scenarios is crucial for solving problems in various fields, from economics to biology.

Series Notation and Formulas

Summation notation interpretation

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• Summation notation uses the Greek letter sigma ($\sum$) to represent the sum of a series of terms
  • Lower limit of summation written below $\sum$ symbol, upper limit written above
  • Expression after $\sum$ symbol represents general term of series
• Arithmetic series: $\sum_{i=1}^{n} (a + (i-1)d)$
  • $a$ first term, $d$ common difference, $n$ number of terms
• Geometric series: $\sum_{i=0}^{n-1} ar^i$
  • $a$ first term, $r$ common ratio, $n$ number of terms

Arithmetic series sum calculation

• Sum of finite arithmetic series calculated using formula: $S_n = \frac{n}{2}(2a + (n-1)d)$
  • $S_n$ sum of first $n$ terms
  • $a$ first term
  • $d$ common difference between consecutive terms
  • $n$ number of terms in series
• Example: Sum of first 10 terms of arithmetic series with $a=2$ and $d=3$
  • $S_{10} = \frac{10}{2}(2(2) + (10-1)3) = 5(4 + 27) = 155$

Geometric series sum computation

• Sum of finite geometric series calculated using formula: $S_n = \frac{a(1-r^n)}{1-r}$, $r \neq 1$
  • $S_n$ sum of first $n$ terms
  • $a$ first term
  • $r$ common ratio between consecutive terms
  • $n$ number of terms in series
• Sum of infinite geometric series calculated using formula: $S_\infty = \frac{a}{1-r}$, $|r| < 1$
  • $S_\infty$ sum of infinite series
  • $a$ first term
  • $r$ common ratio, absolute value must be less than 1 for series to converge
• Example: Sum of infinite geometric series with $a=1$ and $r=\frac{1}{2}$
  • $S_\infty = \frac{1}{1-\frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2$

Sequences and Series

• A sequence is an ordered list of numbers, often represented by a formula
• An infinite series is the sum of all terms in an infinite sequence
• The limit of a sequence or series determines its convergence or divergence
• A recursive formula defines each term of a sequence based on previous terms

Applications of Series

Series concepts in real-world scenarios

• Arithmetic series model situations with constant difference between consecutive terms
  • Regular deposits into savings account
  • Linear growth or decline (population, resources)
• Geometric series model situations with constant ratio between consecutive terms
  • Population growth or decay (bacteria, radioactive material)
  • Compound interest (investments, loans)

Series formulas for financial problems

• Annuity: series of equal payments made at regular intervals
  • Present value of annuity calculated using formula: $PV = PMT \cdot \frac{1-\frac{1}{(1+r)^n}}{r}$
    • $PV$ present value of annuity
    • $PMT$ periodic payment amount
    • $r$ periodic interest rate (decimal form)
    • $n$ total number of payments
• Compound interest: interest calculated on initial principal and accumulated interest from previous periods
  • Future value of lump sum with compound interest calculated using formula: $FV = PV(1+r)^n$
    • $FV$ future value
    • $PV$ present value (initial principal)
    • $r$ periodic interest rate (decimal form)
    • $n$ number of compounding periods
• Example: Future value of $1,000 invested at 5% annual interest, compounded quarterly for 10 years
  • $r = 0.05/4 = 0.0125$ (quarterly interest rate)
  • $n = 4 \cdot 10 = 40$ (number of quarters in 10 years)
  • $FV = 1000(1+0.0125)^{40} \approx$1,643.62