Is the sequence: $\underbrace{2,2,...2}_{1000\text{times}}$ a harmonic sequence? First of all let me give a bit introduction about what is a harmonic sequence. For e.g.: We can say that $1,\frac{1}{2},\frac{1}{3},...\frac{1}{n}$ is a harmonic sequence because if we take the reciprocal of each term of this harmonic sequence and form a new sequence with these reciprocal terms, then that sequence will be in Arithmetic Progression.$(1,2,3,...,n)$ is in Arithmetic Progression.
Now suddenly this question came in my mind that can we say that the sequence:$\underbrace{2, 2,... 2}_{1000\text{times}}$ is in Harmonic Progression? I guess that we can say that the above sequence is in Harmonic Progression. The reason for this is that if we take the reciprocal of each of these terms of this above sequence and form a new sequence with these reciprocal terms then the new sequence will be in Arithmetic Progression.$\underbrace{\frac{1}{2}, \frac{1}{2},... \frac{1}{2}}_{1000\text{times}}$ is an Arithmetic Progression. But also I am having a doubt here.
We know that $$\sum_{i=1}^{1000}2=2000$$. Again we know that the formula of summation of a harmonic sequence is $\ln(n)+\gamma$. So, if I apply this formula here then I am getting the value of the above summation to be
$$\sum_{i=1}^{1000}2=\ln(1000)+0.052$$.(Since $\gamma=0.052$). The value of $(\ln(1000)+0.052)$ is $\approx 6.959$. So I can't understand that why two different values are coming for a summation? Of course, the value of the above summation will be $2000$. But why $2000$ is not coming by applying the formula of summation of Harmonic sequence? Please explain me out.