`sum_(n=0)^oo (n+10)/(10n+1)` Determine the convergence or divergence of the series.

`sum_(n=0)^oo(n+10)/(10n+1)`

For the series `a_n=(n+10)/(10n+1)`

`a_n=(1+10/n)/(10+1/n)`

`lim_(n->oo)a_n=lim_(n->oo)(1+10/n)/(10+1/n)`

`=1/10!=0`

As per the n'th term test for divergence,

If `lim_(n->oo)a_n!=0` , then `sum_(n=1)^ooa_n` diverges

So, the series diverges.

`sum_(n=0)^oo(n+10)/(10n+1)`

For the series `a_n=(n+10)/(10n+1)`

`a_n=(1+10/n)/(10+1/n)`

`lim_(n->oo)a_n=lim_(n->oo)(1+10/n)/(10+1/n)`

`=1/10!=0`

As per the n'th term test for divergence,

If `lim_(n->oo)a_n!=0` , then `sum_(n=1)^ooa_n` diverges

So, the series diverges.