My Favorite Sequences and Series

What is a Sequence or a Series?

It is important to be able to tell them apart in order to avoid confusion

Sequence

A sequence is a list of numbers with an order with a rule

Series

A series is a sum of the terms of a sequence

My favorite Sequences

• Fibonacci Sequence
  0. 0
  1. 1
  2. 1
  3. 2
  4. 3
  5. 5
  6. 8
  7. 13
  8. . . .
• Mersenne primes
  1. 3
  2. 7
  3. 31
  4. 127
  5. 8191
  6. 131071
  7. . . .

My favorite Series

• Taylor expansion of e^x
  1. 1
  2. $+x$
  3. $+\frac{x^2}{\mathrm{2!}}$
  4. $+\frac{x^3}{\mathrm{3!}}$
  5. $+\frac{x^4}{\mathrm{4!}}$
  6. $+\frac{x^5}{\mathrm{5!}}$
  7. . . .
• Geometric Series for ${\left(\frac{1}{2}\right)}^x$
  1. $1$
  2. $+\frac{1}{2}$
  3. $+\frac{1}{4}$
  4. $+\frac{1}{8}$
  5. $+\frac{1}{16}$
  6. $+\frac{1}{32}$
  7. . . .