diff --git a/Grade 10/Math/MCR3U7/Unit 2: Sequences, Series, and Financial Applications.md b/Grade 10/Math/MCR3U7/Unit 2: Sequences, Series, and Financial Applications.md index 708e8e9..e1a4726 100644 --- a/Grade 10/Math/MCR3U7/Unit 2: Sequences, Series, and Financial Applications.md +++ b/Grade 10/Math/MCR3U7/Unit 2: Sequences, Series, and Financial Applications.md @@ -88,4 +88,26 @@ for(int i=1; i<=N; i++) { Either the series **converges** and **diverges**. There is only a finite sum when the series **converges**. -Recall the +Recall the our formula is $`\dfrac{a(r^n-1)}{r-1}`$, and is $`n`$ approaches $`\infty`$, if $`r`$ is less than $`1`$, then $`r^n`$ approaches $`0`$. So this +series converges. Otherwise, $`r^n`$ goes to $`\infty`$, so the series diverges. + +If the series diverges, then the sum can be calculated by the following formula: + +If $`r = \dfrac{1}{2}`$, then $`\large \lim_{x \to \infty} (\frac{1}{2})^x = 0`$ Therefore, $`S_n = \dfrac{a(1 - 0)}{1 - r}`$. This works for any $`|r| \lt 1`$ + +## Binomial Expansion +A binomial is a polynomial expression with 2 terms. + +A binomial expansion takes the form of $`(x + y)^n`$, where $`n`$ is an integer and $`x, y`$ can be any number we want. + +A common relationship of binomial expansion is pascal's triangle. The $`nth`$ row of the triangle correspond to the coefficent of $`(x + y)^n`$ + +``` + 1 row 0 + 1 1 row 1 + 1 2 1 row 2 + 1 3 3 1 row 3 + 1 4 6 4 1 row 4 + 1 5 10 10 5 1 row 5 +``` +