Write the first five decimal approximations by defect and by excess to the number $$e=2,71828182846\ldots$$, as well as the first five fitted intervals of the sequence of intervals that it defines.

### Development:

Given the number $$e=2,71828182846\ldots$$ and considering the rational number obtained by truncation in each of the decimal positions we obtain the sequence of approximation of number $$e$$ by defect:

$$$2; 2.7; 2.71; 2.718; 2.7182; 2.71828; 2.718281; 2.7182818;$$$ $$$2.71828182; 2.718281828; 2.7182818284; 2.71828182846; \ldots $$$

Then, we add a unit in the last digit of every term of this sequence, so we obtain the approximation by excess of the number $$e$$:

$$$3; 2.8; 2.72; 2.719; 2.7183; 2.71829; 2.718282; 2.7182819;$$$ $$$2.71828183; 2.718281829; 2.7182818285; 2.71828182847; \ldots $$$

And from both successions, we can construct the succession of fitted intervals that defines to number $$e=2,71828182846\ldots$$:

$$$[2,3]; [2.7,2.8]; [2.71,2.72]; [2.718,2.719]; [2.7182,2.7183]; [2.71828,2.71829];$$$ $$$[2.718281,2.718282]; [2.7182818,2.7182819]; [2.71828182,2.71828183]; \ldots$$$

### Solution:

The first five terms of the sequence of approximation by defect are: $$2; 2.7; 2.71; 2.718; 2.7182$$ The first five terms of the sequence of approximation by excess are: $$3; 2.8; 2.72; 2.719; 2.7183$$ The first five terms of the sequence of fitted intervals: $$[2,3]; [2.7,2.8]; [2.71,2.72]; [2.718,2.719]; [2.7182,2.7183]$$