Solution 9 (a).

See text and/or instructor's solution manual.

Solution.   Observe that

                      

Change the index of summation in the series and write it as follows

                      

Now use the relation      for    to get  

                    

Thus we have,   .   

Rearrange the terms,   ,   and solve for  .

Therefore,   ,   for all    for some number  R.

We are done.   

Aside.  We can let Mathematica double check our work.

The coefficients    can be defined recursively with the following commands:

A table of the first few values can be displayed:

We are really done.   

Aside.  It is known that the Fibonacci numbers can be represented with the formula:

                    .  

It in interesting to notice that Mathematica can sum the infinite series:

Notice.  Mathematica prefers to write      as   .

We are really really done.   

Resolving the formula for the Fibonicci Numbers.  

It is not too difficult a task to expand    in its partial fraction form:

                    .  

The two terms      and      on the right side of this equation can easily be expanded as geometric series:

                    ,   and

                    .  

Then we can substute these series and obtain

Therefore,  we have shown that  .

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell