use sigma notation to write the maclaurin series for the function ln(1+x)
Since the series is
x - x^2/2 + x^3/3 - x^4/4 + ...
That would be
∞
∑ (-1)^(k-1) x^k/k
k=1
To write the Maclaurin series for the function ln(1+x) using sigma notation, we can start by determining the general term of the series.
The Maclaurin series for ln(1+x) is given by:
ln(1+x) = ∑[ n=0 to ∞ ] [ (-1)^n * (x^n) / n ]
The general term, an, of the series is given by:
an = (-1)^n * (x^n) / n
Now, using sigma notation, we can write the Maclaurin series for ln(1+x) as:
ln(1+x) = ∑[ n=0 to ∞ ] [ (-1)^n * (x^n) / n ]
To use sigma notation to write the Maclaurin series for the function ln(1+x), we need to find the coefficients of the series. The Maclaurin series for ln(1+x) is given by:
ln(1+x) = ∑ [ (-1)^(n+1) * (x^n) / n ]
In this series, we sum up all the terms starting from n=0 to infinity. Each term in the series has the general form [ (-1)^(n+1) * (x^n) / n ].
The key step to finding this series is to recognize that the derivative of ln(1+x) can be expressed as a geometric series. The derivative of ln(1+x) is:
d/dx (ln(1+x)) = 1 / (1+x)
Now, we can integrate both sides of this equation to find the Maclaurin series for ln(1+x):
ln(1+x) = ∫ [ 1 / (1+x) ] dx
Using integral representation for ln(1+x):
ln(1+x) = ∫ [ 1 / (1+x) ] dx
= ∫ [∑ [ (-1)^(n) * (x^n) ]] dx
= ∑ [ (-1)^(n) * ( ∫ (x^n) dx ) ]
= ∑ [ (-1)^(n) * (x^(n+1)) / (n+1) ]
Now, we have expressed ln(1+x) as a series using sigma notation. Each term in the series is given by [ (-1)^(n) * (x^(n+1)) / (n+1) ].
Use sigma notation to write the Maclaurin series for the
function e^-x