Alan

This problem is neatly solved by Presh Talwalker on YouTube. Search for "Mind Your Decisions. The chord progression problem."

Nicely done heureka - a more elegant solution than mine!

For the first one, note that the term 3n⁵/(4n⁵ + 1)  gets very close to 3/4 by the time n has reached 10 or so.  What that means is that the sum consists of an infinite number of values that are all very nearly 3/4.  Adding an infinite number of 3/4 results in an infinite sum.

My result is as follows:

Though there must be a simpler way!

"Coin A is flipped three times and coin B is flipped four times. What is the probability that the number of heads obtained from flipping the two fair coins is the same?"

I get the following:

Edit:  Item number 3 in the list above should be a₂/a₁ not a₂/a₃ !

I get the following:

The two equations are:

-3 - 4t = -4 + 8s    (1)       and   0 + t = k - 2s    (2)

Rearrange these as follows:

from (1):   t = 1/4 - 2s

from (2):   t = k - 2s

So k = 1/4

Hmm!  MathCad gives the following result:

Here's my attempt at #2:

I should note that only the first result (ie a=2, b=6, c=3, d=4) matches tan(7.5°).