i. Show that for any positive integer n the sum of the binomial coefficients ?

n

k ? for even k is the

same as the sum for odd k:

?? n

2 k ? =?? n

2 k + 1 ? .

(For instance, ?

6

6 ? +?

6

4 ? +?

6

2 ? +?

6

0 ? = ?

6

5 ? + ?

6

3 ? +?

6

1 ? ). (You can get this from the binomial expansion

of (1+x)^n for the appropriate value of x.)

ii. Show that the product of the power series for sin(x) and the series for cos(x) is (1/2) times the

power series for sin(2x).

iii. We wish to solve the differential equation y'' - 5 y' + 6 y = 0, with the initial conditions y(0)=2

and y'(0) = 5. There is a general theorem that says there is a unique solution in this case and the solution

has a power series expansion (about 0). Find the power series and then identify the resulting

function as something familiar. (The equation and the initial conditions should enable you to find all of

the derivatives of the solution, and you should then try to identify this as something familiar.)

21. i. Show that for any positive integer n the sum ofthe binomial coefficients ( E ) for even k is the same as the sum for odd k: Z(2=2) (Z)(" 22\") [Forinstance,g( )+(: )4-(3)+(E) =g( )+ (g ) +6 )l. (Youcangetthisfromthebinomialexpan- sion of (1+x]v\ forthe appropriate value of x.) ii. Show that the product of the power series for sinlx) and the series for costs) is (112} times the power series for sinl2x). iii. We wish to solve the differential equation y\" - 5 y' + 6 y = D, with the initial conditions yll=2 and 5'10}: 5. There is a general theorem that says there is a unique solution in this case and the solu- tion has a power series expansion (about 0). Find the power series and then identify the resulting function as something familiar. (The equation and the initial conditions should enable you to find all of the derivatives of the solution, and you should then try to identify this as something familiar.)