Accelerating Convergence by the Euler and Kummer Transformations

Let $S = u_{0} - u_{1} + u_{2} - u_{3} + \ldots$ be a convergent series, where the numbers $u_{0}, u_{1}, u_{2}, \ldots$ are not necessarily positive. Prove the equality:

where $\Delta u_{0}, \Delta u_{1}, \Delta u_{2}, \ldots$ are the differences: $\Delta u_{0} = u_{1} - u_{0}$, $\Delta u_{1} = u_{2} - u_{1}, \ldots$

By repeated application of the method of problem 5269, prove the equality:

where $2^{n}R_{n} = \Delta^{n}u_{0} - \Delta^{n}u_{1} + \Delta^{n}u_{2} - \ldots$

If the numbers $\Delta^{n}u_{0}, \Delta^{n}u_{1}, \Delta^{n}u_{2}, \ldots$ (see problem 5270) have the same sign and decrease in absolute value, then $R_{n} = \dfrac {\theta \cdot \Delta^{n}u_{0}}{2^{n}}$; $\dfrac {1}{2} < \theta < 1$.

Setting $S = u_{0} - u_{1} + u_{2} - u_{3} + \ldots + (-1)^{n}u_{n} + (-1)^{n+1}\rho_{n+1}$, where $\rho_{n+1} = u_{n+1} - u_{n+2} + u_{n+3} - \ldots$, prove the identity

For the quantity $R_{n}$ in problem 5270, prove the equality:

where $\rho_{\nu } = u_{\nu } - u_{\nu +1} + u_{\nu +2} - \ldots$

Prove the theorem: if the series $\displaystyle \sum_{n=0}^{\infty }(-1)^{n}u_{n}$ converges, then

where the series on the right also converges. (Euler's transformation.)

Using the identity $a = \dfrac {a}{1+x} + \dfrac {ax}{1+x}$, prove the identity

where the series $u_{0}x - u_{1}x^{2} + \ldots$ converges by assumption for $-1 < x < 1$.

Under the same conditions (see problem 5275), prove the equality:

where $y = \dfrac {x}{1+x}$; $\Delta u_{0}, \Delta^{2}u_{0}, \Delta^{3}u_{0}, \ldots$ are the successive differences, and the quantity

where $\rho_{p} = u_{p}x^{p} - u_{p+1}x^{p+1} + \ldots$

Prove that

if $0 < x < 1$, and the series on the left converges.

Prove, for $m>0$, $x>-\dfrac {1}{2}$, the equality:

Prove the equality:

Using Euler's transformation, find to within $10^{-5}$ the sum of the series $\ln 2 = 1 - \dfrac {1}{2} + \dfrac {1}{3} - \dfrac {1}{4} + \ldots$ (It is useful to sum the first several terms directly.)

Using Euler's transformation, find to within $10^{-5}$ the sum of the series $\dfrac {\pi }{4} = 1 - \dfrac {1}{3} + \dfrac {1}{5} - \dfrac {1}{7} + \ldots$ (It is useful to sum the first several terms directly.)

Using Euler's transformation, find to within $10^{-5}$ the sum of the series $1 - \dfrac {1}{3^{2}} + \dfrac {1}{5^{2}} - \dfrac {1}{7^{2}} + \ldots$ (It is useful to sum the first several terms directly.)

Using Euler's transformation, find to within $10^{-5}$ the sum of the series $1 - \dfrac {1}{2^{3}} + \dfrac {1}{3^{3}} - \dfrac {1}{4^{3}} + \ldots$ (It is useful to sum the first several terms directly.)

Using Euler's transformation, find to within $10^{-5}$ the sum of the series $\dfrac {1}{1\cdot 2\cdot 3} - \dfrac {1}{2\cdot 3\cdot 4} + \dfrac {1}{3\cdot 4\cdot 5} - \ldots$ (It is useful to sum the first several terms directly.)

Using Euler's transformation, find to within $10^{-5}$ the sum of the series $\dfrac {1}{\ln 10} - \dfrac {1}{\ln 11} + \dfrac {1}{\ln 12} - \dfrac {1}{\ln 13} + \ldots$ (It is useful to sum the first several terms directly.)

Prove that applying Euler's transformation to the series $\displaystyle \sum_{n=0}^{\infty }\left(-\frac{1}{2}\right)^{n}$ improves the convergence, while applying it to the series $\displaystyle \sum_{n=0}^{\infty }\left(-\frac{1}{4}\right)^{n}$ worsens it.

Setting, for the series $\sum \dfrac {1}{n^{2}}$, $C_{n} = \dfrac {1}{n^{2} - \frac14}$, prove the equality:

Prove the equality:

From the series $\dfrac {\pi^{2}}{6} = \displaystyle \sum_{n=1}^{\infty } \dfrac {1}{n^{2}}$, choosing $p_{n}$ in the form $an+b+\dfrac {c}{n+d}$, where $a,b,c,d$ are appropriate coefficients, obtain by Kummer's transformation the equality:

From the series $\dfrac {\pi^{2}}{8} = \displaystyle \sum_{n=0}^{\infty } \dfrac {1}{(2n+1)^{2}}$, obtain by Kummer's transformation the formula:

Prove the equality:

Prove the equality:

Prove the equality:

Prove the equality:

Prove the equality:

Prove the equality:

Prove the equality:

where $u_{n} = (-1)^{n}\left(\frac{1\cdot 3\cdot 5\ldots (2n-1)}{2\cdot 4\cdot 6\ldots 2n}\right)^{2}$.

Kummer's transformation in its simplest form is usefully applied to trigonometric series. Applying a transformation of this type, prove, for $0<x<\pi$, the equality:

Applying Kummer's transformation, prove, for $0<x<\pi$, the equality:

Applying Kummer's transformation, prove, for $0<x<\pi$, the equality:

Applying Kummer's transformation, prove, for $0<x<\pi$, the equality:

Prove, for $0<x<\pi$, the equality:

Prove that the function $y = \displaystyle \sum_{n=1}^{\infty } \frac{\sin nx}{n^{2}+1}$, for $0<x<\pi$, satisfies the differential equation

Prove that the function $y = \displaystyle \sum_{n=1}^{\infty } \frac{n^{2}}{n^{3}+1}\sin 2nx$, for $0<x<\dfrac {\pi }{2}$, satisfies the equation