It's simple! Just add it!

Here's a solution written in python 2.x

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>>> import math
>>> sum([1.0/(i**(math.sqrt(i))) for i in xrange(1, 16)])
1.6379964989896623

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1
1.375214227
1.524356839
1.586856839
1.614213026
1.626627448
1.632436172
1.635226632
1.636598374
1.637286586
1.63763822
1.637820865
1.637917163
1.637968637
1.637996499

Answer： $\boxed{6}$

Python one-liner -

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sum([1.0/(x**(x**0.5)) for x in xrange(1, 16)])

While the summation seems to be challenging when done by hand, its corresponding code is incredibly short.

Below is a code in C.

#include    <stdio.h>
#include    <math.h>

double function(int x)
{
    return(1.0/pow((double)x, pow((double)x, 0.5))) ;
}

int main(void)
{
    double sum = 0.0 ;
    int x ;

    for(x = 1; x <= 15; x++)
        sum += function(x) ;

    printf("%lf", sum) ;

    return 0 ;
}

When the program is compiled and run, the output will be

1.637996

So the first digit after the decimal is 6 .

Python 3.4

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def g(n):
    return n**(-1*(n**0.5))
a=0
for i in range(1,16):
    a+=g(i)
print(a)