# Problem Proposed for the American Mathematical Monthly

### Description

The problem is to define: P(x) := {Sigma}{sub k = 1}{sup {infinity}} arctan (x - 1/(k + x + 1) {radical}(k + 1) + (k + 2) {radical}(k + x)). (1) (a) Find explicit, finite-expression evaluations of P(n) for all integers n {ge} 0. (b) Show {tau} := lim{sub x {yields} -1{sup +}} P(x) exists, and find an explicit evaluation for {tau}. (c) Are there a more general closed forms for P, say at half-integers? Solution with the abbreviations: r := {radical} (k + 1), s := {radical} (k + x) the argument of arctan in (1) becomes s{sup 2} ... continued below

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### Creation Information

Bailey, David H.; Borwein, Jonathan M. & Waldvogel, Jorg November 14, 2008.

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## What

### Description

The problem is to define: P(x) := {Sigma}{sub k = 1}{sup {infinity}} arctan (x - 1/(k + x + 1) {radical}(k + 1) + (k + 2) {radical}(k + x)). (1) (a) Find explicit, finite-expression evaluations of P(n) for all integers n {ge} 0. (b) Show {tau} := lim{sub x {yields} -1{sup +}} P(x) exists, and find an explicit evaluation for {tau}. (c) Are there a more general closed forms for P, say at half-integers? Solution with the abbreviations: r := {radical} (k + 1), s := {radical} (k + x) the argument of arctan in (1) becomes s{sup 2} - r{sup 2}/(s{sup 2} + 1) r + (r{sup 2} + 1) s = s - r/r s + 1 = 1/r - 1/s / 1 + 1/r 1/s.

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### Source

• Journal Name: American Mathematical Monthly

### Identifier

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• Report No.: LBNL-2137E
• Grant Number: DE-AC02-05CH11231
• Office of Scientific & Technical Information Report Number: 963543

### Collections

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## When

### Creation Date

• November 14, 2008

### Added to The UNT Digital Library

• Nov. 13, 2016, 7:26 p.m.

### Description Last Updated

• Nov. 18, 2016, 4:12 p.m.

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