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Problems #5,6 are graded.
Problem 5: Let {ak}∞k=0={13k}∞k=0 and {bk}∞k=0={17k}∞k=0. Then compute using geometric series:
⟨{ak},{bk}⟩=∞∑k=013k17k=∞∑k=0(121)k=11−121=2120.
Now let {ck}∞k=0={√1k!}. Using the fact that ex=∞∑k=0xkk! we see that e1=e=∞∑k=01k!. Now we may compute
⟨{√1k!},{√1k!}⟩=∞∑k=0√1k!√1k!=∞∑k=01k!=e.
Problem 6: Let →x=5+4i and →y=21+16i. Compute
⟨→x,→y⟩=(5+4i)¯(9−11i)=(5+4i)(9+11i)=45+55i+36i+44i2=45+91i−44=1+91i.
Let z1=21+16i and notice
z2=11−5i2+i=11−5i2+i2−i2−i=22−11i−10i+5i24−i2=17−21i5=175−215i.
Now compute
⟨z1,z2⟩=(21+16i)¯(175−215i)=(21+16i)(175+215i)=3575+4415i+2725i+3365i2=215+7135i.
