Obtain precise numerical values of an expanded Harmonic number?

Questions
1

Thank you for opening this thread. I am a university student.

Could anyone help me to calculate the following summation numerically under a certain precision?

fig0.png

This is an expansion of Harmonic Number by introducing the parameter m. Setting m=1 implies

HarmonicNumber[n].

PROBLEM:

If n is not so big, e.g., n<=40, I can obtain the correct results with m=10/9 (see, Figure 1).

But when n goes to large, the numerical results behave not stable (see, Figure 2).

fig1.png
fig1.png523×375 3.64 KB

fig2.png
fig2.png523×354 3.12 KB

How can I avoid this abnormal results technically? Actually I need to calculate the case of n=100 and m>1.

Thank you in advance.

2

I did not try, because you did submit any copy-paste-able code:

but try adding

 WorkingPrecision -> 50

at the end of your plot command.

3

The imprecision may be due to floating-point arithmetic. You can switch to arbitrary precision arithmetic by giving N with an explicit number of digits for the output:

DiscretePlot[N[s[10/9, n], 16], {n, 40, 80}]
4

Thank you for your replies.

Actually DiscretePlot with WorkingPrecision->50 did worked.

To show the value, I used the N operator suggested by Gianluca.

Thank you!