1. Calculate
and
to two places of decimals by means of the power series for
and
.
2. Prove that and .
3. Prove that if then and .
4. Since we have Prove by multiplying the two series on the right-hand side (§ 195) and equating coefficients (§ 194) that Verify the result by means of the binomial theorem. Derive similar identities from the equations
5. Show that
6. Expand in powers of . [We have and similarly Hence
7. Expand , , and in powers of .
8. Expand and in powers of . [Use the formulae It is clear that the same method may be used to expand and , where is any integer.]
9. Sum the series
[Here
and similarly
Hence
10. Sum
11. Sum and the corresponding series involving sines.
12. Show that
13. Show that the expansions of and in powers of (Ex. LVI. 1) are valid for all values of and , real or complex.