Core Pure 1 Chapter 3: Series

7 flashcards · Mathematics · by TheoTyson

1. What is the formula for the sum of a series with constant terms?

   ∑r = 1n1=n\displaystyle \sum_{r \, = \, 1}^n 1 = nr=1∑n​1=n

2. What is the formula for the sum of the first nnn natural numbers?

   ∑r = 1nr=12n (n + 1)\displaystyle \sum_{r \, = \, 1}^n r = \frac{1}{2} n \, (n \, + \, 1)r=1∑n​r=21​n(n+1)

3. What is the formula for the sum of the squares of the first nnn natural numbers?

   ∑r = 1nr2=16n (n + 1) (2n + 1)\displaystyle \sum_{r \, = \, 1}^n r^2 = \frac{1}{6} n \, (n \, + \, 1) \, (2n \, + \, 1)r=1∑n​r2=61​n(n+1)(2n+1)

4. What is the formula for the sum of the cubes of the first nnn natural numbers?

   ∑r = 1nr3=14n2 (n + 1)2\displaystyle \sum_{r \, = \, 1}^n r^3 = \frac{1}{4} n^2 \, (n \, + \, 1)^2 r=1∑n​r3=41​n2(n+1)2

5. How do you work out the sum of a series that does not start at r=1r=1r=1 ?

   ∑r = knf(r)=∑r = 1nf(r)−∑r = 1k−1f(r)\displaystyle \sum_{r \, = \, k}^n f(r) = \displaystyle \sum_{r \, = \, 1}^n f(r) - \displaystyle \sum_{r \, = \, 1}^{k-1} f(r)r=k∑n​f(r)=r=1∑n​f(r)−r=1∑k−1​f(r)

6. Rearrange ∑r = 1nkf(r)\displaystyle \sum_{r \, = \, 1}^n kf(r)r=1∑n​kf(r)

   ∑r = 1nkf(r)=k∑r = 1nf(r)\displaystyle \sum_{r \, = \, 1}^n kf(r) = k \displaystyle \sum_{r \, = \, 1}^n f(r)r=1∑n​kf(r)=kr=1∑n​f(r)

7. Rearrange ∑r = 1n(f(r) + g(r))\displaystyle \sum_{r \, = \, 1}^n (f(r) \, + \, g(r)) r=1∑n​(f(r)+g(r))

   ∑r = 1n(f(r) + g(r))=∑r = 1nf(r)+∑r = 1ng(r)\displaystyle \sum_{r \, = \, 1}^n (f(r) \, + \, g(r)) = \displaystyle \sum_{r \, = \, 1}^n f(r) + \displaystyle \sum_{r \, = \, 1}^n g(r)r=1∑n​(f(r)+g(r))=r=1∑n​f(r)+r=1∑n​g(r)