# MS 095 Fractional Series

Suppose I walk half way towards a wall. And then I walk half of the remaining half. And then half of the half of the remaining half... how far have I walked across the room after each "half"?

The extended family of problems that include the problem are called series. Mathematically we can work out the answer in the following way:

Half way to the wall:

And then half of remaining half:

Then half of the half of the remaining half:

Calculate the following, writing the answer as both a fraction and decimal:

= ______________(decimal form)

= ______________(decimal form)

= ______________(decimal form)

= ______________(decimal form)

Is there a pattern in the fractions? _______

What will be the fractional answer for the next sum in the series? ______

What is the decimal equivalent for the fraction above? _____

The pattern can also be written:

In advanced mathematics this is written using the following shorthand:

If n=1 then the result is ½ or 0.5

If n=2 then the result is ¾ or 0.75

If n=3 then the result is 7/8 or 0.875... the same pattern of results as above.

Find the fractional and decimal result for n=10:

Fraction:

Decimal:

Does the series appear to be adding up to a particular number? ____

What number does this series appear to be getting closer and closer to? _____

If a series gets closer and closer to a number without passing that number the series is said to converge on that number.

To what number does it appear the series above is converging? ____

In advanced math we would write the above answer after the sum:

= ____

The above problem is part of a family of related problems. Calculate the following in decimal form:

= ___ = ___ = ___ = ______

= ______= ______

To what does it appear this series might be adding up to? _____

= _____

Calculate the following in decimal form:

= ___ = ___ = ___ = ______

= ______= ______

To what does it appear this series might be adding up to (converge on)? _____

= _____

What number will converge on? _____

Big toughie: What will converge on? _____