Doesthe infiniteseries1 + 0.1 + 0.01 + 0.001+ ... convergeor diverge?
Solution
To makeour workeasier, writethe infiniteseries1 + 0.1 + 0.01 + 0.001+ ... as an infiniteseriesof fractions:
To solvefor convergenceor divergenceof the infiniteseries,writethe formulafor thenth partialsum
. Notethat thenth partialsum endswith a
powerofn – 1in the denominatorbecause1 is the first term of the infiniteseries.
It is ratherdifficult to find as it is written.We will
“work”the sum into a differentform so that we can find the limit of the sequenceof partialsums.
First,multiplyboth sidesof the equation by :
Now we havetwo equations:
Subtractthe bottomequationfrom the top equationto canceltermsand simplifying: