CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 12. Discrete Math

3. Just like summation uses a capital Greek letter forS, product uses a capital Greek letter forPwhich is the capital
   form ofπ.

1 ·sin(^3602 · 3 )·sin(^3602 · 4 )·sin(^3602 · 5 )·sin(^3602 · 6 )·sin(^3602 · 7 )·...=

∞

i∏= 3 sin

( 360

2 ·i

)

This infinite product is the result of starting with a circle of radius 1 and inscribing a regular triangle inside the circle.
Then you inscribe a circle inside the triangle and a square inside the new circle. The shapes alternate being inscribed
within each other as they are nested inwards: circle, triangle, circle, square, circle, pentagon, ... The question that
this calculation starts to answer is whether this process reduces to a number or to zero.

Practice

For 1-5, write out all the terms of the sigma notation and then calculate the sum.

1. ∑^5
   k= 1

2 k− 3

2. ∑^8
   k= 0

2 k

3.i∑=^412 · 3 i

10

i∑= 14 i−^1

4

i∑= 02 ·

( 1

3

)i

Represent the following series in summation notation with a lower index of 0.

6. 1+ 4 + 7 + 10 + 13 + 16 + 19 + 22


7. 3+ 5 + 7 + 9 + 11


8. 8+ 7 + 6 + 5 + 4 + 3 + 2 + 1


9. 5+ 6 + 7 + 8


10. 3+ 6 + 12 + 24 + 48 +···


11. 10+ 5 +^52 +^54


12. 4− 8 + 16 − 32 + 64 ···


13. 2+ 4 + 6 + 8 +···
    14.^13 +^19 + 271 + 811 +···
    15.^23 +^29 + 272 + 812 +···