Limits at Infinity
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Help:
(x+y)1 = x + y
(x+y)2 = x2 + 2xy + y2
(x+y)3 = x3 + 3x2y + 3xy2 + y3
(x+y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4
(x+y)5 = x5 + 5x4y + 10x3y2 +10x2y3 + 5xy4 + y5
There are several things that you hopefully have noticed after looking at the expansion
Example:
Let a = x, b = 2, n = 5 and substitute. (Do not substitute a value fork.) |
Geometric Partial Sum
~Take the first equation and multiply it by (r) to get the second equation
~ Subtract the two equations, cancel out like terms to end with:
~ Take out an Sn out from the left side and a1 from the right side so you end up with:
~divide both sides by (1-r) to get the final equation:
EX.) Let an be geometric sequence with a3 = 12 and a6 = 96. Find the 10th partial sum.
To find (r), first find the difference between n;
a6-a3= 3
3 becomes the power of r which equals the fraction of two terms.
take the cube root of 8
r= 2
Plug numbers into the partial sum formula
r= 2
a1= 3
n=10
Infinite Sum