#1 + 1/7 + 1/49# is a geometric series.
the general term for geometric sequences is #u_n = ar^(n-1)#.
#r# is the common ratio - the number that one number in the sequence is multiplied by to get to the next one.
here, the common ratio is #1/7 div 1#, which is the same as #1/49 div 1/7#.
both are equal to #1/7#, so the common ratio #r# is #1/7#.
#a# is the starting number in the sequence - here, this is #1#.
with the values #1# and #1/7# for #a# and #r#, you can find the infinite sum with this formula:
#S_oo = a/(1-r)#
NB: this only works if #r# is between the numbers #-1# and #1#. here, #r = 1/7#, so the formula can be used.
#a/(1-r) = 1/ (1 - 1/7)#
#1 - 1/7 = 6/7#
#1/(6/7)# is the same as #7/6#.
this means that the infinite sum of the sequence where #u_n = (1/7)^(n-1)# is #7/6#.
