I can't tell you why the answers are different, without explanation
I cannot follow what you think you were doing.
Also given that my answere is cprrect to 10 digits, and yours in supposedly
analyticaly derived being close in wjhatever sense is meaningless. the answers
are different, when they should agree to within the precision of the calculations.
RonL
OK with some work I can make some sense of what you have written.
You have:
ln(1+x) = sum_{k=0 to infty} (-1)^k x^{k+1}/(k+1),
so:
(1/x)ln(1+x/5) = sum_{k=0 to infty} (1/x) (-1)^k (x/5)^{k+1}/(k+1),
and you proceed to integrate term by term to get:
integral_{x=0 to 1} (1/x)ln(1+x/5)
.................. = sum_(k=0 to infty) (-1)^k x^{k+1}/[5^{k+1}(k+1)(k+1)] |_0^1.
.................. = sum_(k=0 to infty) (-1)^k/[5^{k+1}(k+1)(k+1)]
lets assume this is correct.
How do you evaluate this:
sum_(k=0 to infty) (-1)^k/[5^{k+1}(k+1)(k+1)]
infinite sum which is the integral you require?
OK lets have a go at evaluating this sum:
sum_(k=0 to infty) (-1)^k/[5^{k+1}(k+1)^2]
........... = 1/[5] - 1/[5^2*2^2] + 1/[5^3*3^2] - ..
........... ~= 0.190889
Now the error in this is of the same order as the first ignored term or ~-0.0001,
so your integral is 0.191 correct to 3 significant digits, which agrees with my
calculation.
RonL