MaplePrimes - answers and comments on Question, Unable to solve the following inequality.

fsolve

ThU — Sun, 04 Nov 2012 12:52:34 Z

This has to be done numerically, so use fsolve. To get a good initial starting point for the numerical solver, make a plot.

restart:
f:=sum(1/n, n = 1 .. N);
plot([f,20.12],N=1..1e9,y=20..21,gridlines=true);
fsolve(sum(1/n, n = 1 .. N) = 20.12, N=3*10^8);

Axel Vogt — Sun, 04 Nov 2012 20:20:33 Z

As ThU suggests: plotting is a good idea to get an impression.

And you see, that the graph seems to become quite flat for a region you
may need for a possible solution, it looks like a logarithm.

Using the command 'asympt' you can even verify, that this is the case.

Now switch to exp( fct ) = exp (solution ) and plotting will tell you, that
the problem now is (almost) linear.

For such a situation fsolve has no problems at all and works without
an initial guess

Kitonum — Mon, 05 Nov 2012 09:35:51 Z

First, we isolate the N and then find it by bisection of the interval.

restart;

f:=N->sum(1/n, n=1..N):

for k from 0 do

if evalf(f(2^k))>20.12 and evalf(f(2^(k-1)))<20.12 then break: fi:

od:

a:=2^(k-1): b:=2^k:

print([a, b]); # isolation interval

while b-a>=1 do

c:=(a+b)/2:

if (evalf(f(a))-20.12)*(evalf(f(c))-20.12)<=0 then b:=c else a:=c fi:

od:

if evalf[20](f(ceil(b)-1))<=20.12 then print(N=ceil(b)) else

print(N=ceil(b)-1); fi;

[268435456, 536870912]

N = 307130819