Exercise.

(a) Explain why, when proving the convergence of a series
=0
_{
}
it is permissible to ignore
finitely many terms of the series.

(b) Use (a) together with a form of the comparison test to show
that the series
=0
1(2
-5)(2
-3)(2
-3)
converges.

## Solution to selected parts of some of these exercises

### Exercise 1(a)

Let _{
=
=0
} be the th partial
sum. The series
=0
_{
}
converges iff the sequence (_{
)
} of partial sums converges.

Now suppose that
_{
=
=
}
is the partial sum formed by ignoring the first entries
and
_{
}
as
.
Then _{
=0+1++
-1+
}
and _{
0+1++
-1+
}
by continuity of the operation of adding the constant value
_{0+}_{1++}_{
-1
}
. So the
original series also converges.