Answer these 3 questions the 3 and 5 questions must have sigma notation.

3. Find Taylor series for f(x) = = , at a = 3 X 4. Find Maclaurin series for cos(x ) using the maclaurin series for sin (x). 1 5. Find the Binomial Series for (1-x)Example 1: Find the Binomial series for ( 1 + x ) 1 Formula: (1 + x)? = (1+ x)-2 (1 + x)*= 1 + kx + *-1) *2 - Kik-1316-2) 3! 2x3+ ... Solution: (1 + x)-2= 1 - 2x + -2(-2-1) 1 x2 4 -2(-2-1)(-2-2) 2! x3 + ... 3! (1 + x)-2 = 1 - 2x + 3x2 - 4x3+ ... In sigma notation, or ( +1) x" (-1)" n=ll Example 2: Find the Binomial series for v1 + x - (1 + x)1/2 Formula: (1 +x)- 1+ kx+MA-1 2 , Milk-2 a MA- 14-24-2- Solution: (1 + x)1/2= 1+ =x+ -1) 2: x2 + 3! 4: =1+=*+ 2! 3! 1! 1 1 . 1 1 . 1 . 3 1 . 1 . 3 .5 = 1+ 21 . 1!' X -x2 . x3 22 . 2! 23 . 3! 24 - 4! x4+... In sigma notation =1+ 2 + (-1)741[1 .3.5.. (2n - 3)] 2" . 12